Input:
**User**:
What is the smallest positive multiple of 450 whose digits are all zeroes and ones?
Please reason step by step, and put your final answer within \boxed{}.

<think>
Okay, so I need to find the smallest positive multiple of 450 where all the digits are either 0 or 1. Let me start by understanding the problem better.

First, 450 is the number we're dealing with. Let me factorize it to see what we're working with. 450 can be broken down into prime factors. Let me do that:

450 divided by 2 is 225. Then 225 divided by 3 is 75, divided by 3 again is 25, and then divided by 5 is 5, and divided by 5 again is 1. So the prime factors are 2 × 3² × 5². So 450 = 2 × 3² × 5².

Now, since we need a multiple of 450, the number we're looking for must be divisible by 2, 9, and 25. Let me recall the divisibility rules for these numbers.

Divisibility by 2: The number must end with an even digit. Since our digits can only be 0 or 1, the last digit has to be 0. So whatever number we come up with must end with a 0.

Divisibility by 25: A number is divisible by 25 if its last two digits are 00, 25, 50, or 75. But since our digits can only be 0 or 1, the only possible ending is 00. Therefore, the number must end with two zeros.

So combining these two, the number must end with at least two zeros. Also, since it's divisible by 2, which is already covered by the two zeros.

Now, divisibility by 9: A number is divisible by 9 if the sum of its digits is a multiple of 9. Since all digits are 0 or 1, the sum of the digits is just the number of 1s in the number. Therefore, the number of 1s must be a multiple of 9.

So putting it all together, the number must:

1. End with two zeros.
2. Have a number of 1s that is a multiple of 9.
3. Be composed only of 0s and 1s.

Our task is to find the smallest such number.

So, the problem reduces to finding the smallest number made up of 0s and 1s, ending with two zeros, and with the number of 1s being a multiple of 9.

Since we want the smallest number, we need the number with the least number of digits possible. Let me think about how to approach this.

First, since the number must end with two zeros, we can think of the number as something like [some combination of 0s and 1s] followed by '00'.

So, the number can be represented as N = X * 100, where X is a number made up of 0s and 1s.

Therefore, N must be divisible by 450, which is 2*3²*5². Since N is X*100, and 100 is 2²*5², so N = X * 2² * 5². Therefore, for N to be divisible by 450 = 2*3²*5², we need X to be divisible by 3² = 9.

Therefore, X must be a number made up of 0s and 1s, and divisible by 9.

So, the problem simplifies to finding the smallest number X composed of 0s and 1s that is divisible by 9, and then appending two zeros to it.

Therefore, the original problem reduces to finding the minimal X (with digits 0 and 1) divisible by 9, then multiplying by 100.

So, now the problem is: find the smallest number made up of 0s and 1s divisible by 9.

Once we find that X, then N = X * 100 is our answer.

So, how do we find the smallest number made up of 0s and 1s divisible by 9?

Well, since divisibility by 9 requires that the sum of digits is a multiple of 9. Since digits are 0s and 1s, the sum is equal to the number of 1s. Therefore, the number of 1s must be a multiple of 9.

Therefore, the minimal number of 1s is 9. Therefore, the minimal number would be the smallest number with 9 ones and any number of zeros, arranged in such a way that the number is as small as possible.

But how do we arrange the digits? To make the number as small as possible, we need the number to have the least number of digits possible, and within that, the digits should be arranged in ascending order (i.e., leading digits as small as possible).

But since we need the number to be as small as possible, we need the minimal number of digits. The minimal number of digits is 9 (all ones). However, the number 111111111 (nine 1s) is 111,111,111. However, maybe there is a smaller number with more digits but with zeros in between? Wait, no. Because adding zeros would make the number longer, which would make it larger. For example, 101111111 is longer than 111111111? Wait, no. Wait, 101111111 has 9 digits as well. Wait, same number of digits.

Wait, but numbers with more digits are larger. For example, 100000000 is 100 million, which is larger than 111,111,111? Wait, no. Wait, 100,000,000 is 100 million, which is less than 111,111,111. Wait, so actually, numbers with more zeros can be smaller? Wait, but if we have the same number of digits, then the number with more zeros would be smaller.

Wait, but if we have a number with 9 digits, the minimal number would be 100000000, but that has only one 1. However, we need the number of 1s to be 9. Therefore, the minimal number with 9 ones is 111111111. However, if we can have a number with more digits but with some zeros, but with the same number of 1s, but arranged in such a way that the number is smaller.

Wait, for example, 101010101 is a 9-digit number with 5 ones. But we need 9 ones. So, if we have 9 ones, the minimal number would be the one with the ones as far to the right as possible? Wait, no. Wait, to make the number as small as possible, we need the leftmost digits to be as small as possible. So, the first digit should be 1 (since leading zeros are not allowed), then the next digits should be 0s as much as possible.

Wait, but if we have 9 ones, the minimal number would be 100000000... but that only has one 1. Wait, no. Wait, we need exactly 9 ones. So, the minimal number would be the one with the 9 ones placed in the positions that make the number as small as possible.

For example, the smallest number with 9 ones is 111111111 (nine 1s). However, if we can have a number with more digits but with some zeros, but still having 9 ones, but being smaller.

Wait, for example, 1000000001 has 10 digits, with two ones. But we need nine ones. So, 10000000011 would have 10 digits with three ones. Wait, but we need nine ones. So, for example, 100000000... with nine ones. Wait, but how?

Wait, if we have a number with more digits, but with nine ones, but arranged so that the number is smaller. For example, 100000000... with the ones spread out.

But actually, the number with the least value would be the one with the ones as far to the right as possible. For example, if you have a number with 9 ones, the smallest number is 111111111 (nine digits). If you have a number with ten digits, the smallest number with nine ones would be 1000000001, but that's actually larger than 111111111. Wait, 1000000001 is 1,000,000,001, which is larger than 111,111,111.

Wait, so actually, the minimal number with nine ones is 111,111,111. However, maybe there is a smaller number with more digits? Wait, no. Because numbers with more digits are larger. For example, 10 digits would be larger than 9 digits. Therefore, the minimal number is 111,111,111.

But wait, let me check. Suppose we have a number with 18 digits, but with nine ones. But that would be way larger. So, no. Therefore, the minimal number is 111,111,111.

But wait, hold on. Let me think again. Suppose we have a number with 9 ones and some zeros. For example, 1011111111. Wait, but that's 10 digits. But 1011111111 is 1,011,111,111, which is larger than 111,111,111.

Alternatively, if we have 111111111, which is 111,111,111. So, that's smaller.

But maybe there is a number with fewer digits? Wait, no. Because to have nine ones, you need at least nine digits. So, the minimal number of digits is nine. Therefore, the minimal number is 111,111,111.

But wait, is there a smaller number with nine ones? For example, if we have leading zeros? No, because numbers can't have leading zeros. So, the first digit has to be 1. Then, the rest can be arranged.

But if we have nine digits, all ones, that's the minimal. However, maybe there is a number with more digits but with some zeros that is smaller? For example, 100000000... but with nine ones. Wait, but if you have more digits, you need to have more digits. For example, if you have ten digits, you need to have nine ones and one zero. The smallest such number would be 1000000001, but that's 1,000,000,001, which is larger than 111,111,111.

Therefore, the minimal number is 111,111,111.

But wait, let me check if there is a smaller number with nine ones. For example, 101010101... but that has five ones. Not nine.

Alternatively, maybe arranging the ones in a way that the number is smaller. For example, 1001111111. Wait, that's ten digits. Let me check the value. 1,001,111,111. Which is still larger than 111,111,111.

Alternatively, 111,111,111 is 111 million, while 1,001 million is 1.001 billion, which is way larger.

Therefore, it seems that 111,111,111 is the minimal number with nine ones.

But wait, hold on. Let me think again. Maybe there is a number with fewer digits? Wait, no. Because to have nine ones, you need at least nine digits. So, nine digits is the minimum.

But then, if X is 111,111,111, then N = X * 100 = 111,111,111 * 100 = 11,111,111,100.

But is this the minimal? Wait, maybe there is a smaller X?

Wait, but X has to be a number made up of 0s and 1s, divisible by 9. So, maybe there is a smaller number than 111,111,111?

Wait, for example, maybe a number with more digits but with some zeros, which is smaller?

Wait, for example, let's think of numbers with digits 0 and 1, divisible by 9.

The sum of digits must be 9, 18, 27, etc.

So, the minimal sum is 9. Therefore, the minimal number of digits is 9. But maybe there is a number with more digits but with sum 9, but smaller in value?

For example, 1000000001 has sum 2, not 9.

Wait, if we have a number with sum 9, but with more digits. For example, 100000000... with nine ones. But how?

Wait, for example, 100000000111111111. Wait, that's 1 followed by eight zeros and then eight ones? Wait, no. Let me think.

Wait, if I have a number with ten digits, with nine ones. Then, the number would be 1111111111 (ten ones) but that's sum 10. Not 9.

Wait, to have sum 9 with ten digits, we need nine ones and one zero. The smallest such number would be 1011111111. Which is 1,011,111,111. Which is larger than 111,111,111.

Similarly, with eleven digits, the smallest number with nine ones would be 10011111111, which is 10,011,111,111, which is way larger.

Therefore, the minimal number is indeed 111,111,111.

But wait, let me check if there is a number with sum 9, but with digits arranged in a way that it's smaller. For example, maybe 101111111? Wait, that's nine digits. Let me check the sum: 1 + 0 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 1 + 0 + 7*1 = 8. Not 9.

Wait, so to get sum 9, we need nine 1s. Therefore, the number must have nine 1s. Therefore, the minimal number is 111,111,111.

But wait, maybe there is a number with more digits but with some zeros that is smaller? For example, 100000000... but with nine ones. Wait, but how? If you have more digits, you need to have more digits. For example, if you have ten digits, you need nine ones and one zero. The smallest number would be 1000000001, but that has only two ones. Wait, no. Wait, if you have ten digits with nine ones, then you have one zero. The smallest number would be 1000000001? No, that's 1 followed by eight zeros and then a 1. Wait, but that's ten digits. Wait, the number would be 1000000001, which is 1 followed by eight zeros and then a 1. But that has two ones. Wait, no. Wait, if you have nine ones and one zero, then the number would be 1000000001 (with one zero and nine ones?), no. Wait, no. Wait, ten digits: one zero and nine ones. So, the number would be 1111111110, which is 111,111,1110. Wait, that's 1,111,111,110. Which is larger than 111,111,111.

Alternatively, if the zero is in the middle, like 1111111101, which is 1,111,111,101. Still larger.

Therefore, the minimal number with nine ones is 111,111,111.

Therefore, X is 111,111,111, and N is 111,111,111 * 100 = 11,111,111,100.

But wait, is this the minimal? Let me check if there is a smaller X.

Wait, maybe there is a number with sum of digits 9, but with more digits, but smaller in value? For example, maybe 1001001001... but let me check.

Wait, for example, 1001001001 has sum 4. Not 9.

Alternatively, 1110000000 has sum 3. Not 9.

Alternatively, 111111111 is the only nine-digit number with sum 9.

But maybe there is a number with more digits but with sum 9, but smaller?

Wait, for example, 100000000111111111 (which is 1 followed by eight zeros and then eight ones). Wait, that's 17 digits. But the value is 100,000,000,111,111,111, which is way larger than 111,111,111.

Therefore, no.

Alternatively, maybe there is a number with sum 9, but with digits arranged in a way that it's smaller. For example, 101010101... but with more ones.

Wait, for example, 101010101 has five ones. If I add more ones... but how?

Wait, if I have 1011111111, which is ten digits, with eight ones? Wait, no. Let me count. 1, 0, 1, 1, 1, 1, 1, 1, 1, 1. That's 1 + 0 + 7*1 = 8. Not 9.

Wait, to get nine ones, we need nine 1s. So, for example, 1011111111 has nine ones? Let me count: 1, 0, 1, 1, 1, 1, 1, 1, 1, 1. That's 1 + 0 + 8*1 = 9. Yes! So that's nine ones.

So, the number is 1011111111. Which is 1,011,111,111.

Compare that to 111,111,111. Which is smaller?

Well, 111,111,111 is 111 million, while 1,011,111,111 is 1.011 billion. So, 111 million is smaller.

Therefore, 111,111,111 is smaller.

But wait, is there a number with more digits but smaller? For example, 1001111111. Let me check.

That's 1,001,111,111. Which is 1.001 billion, which is larger than 111 million.

Alternatively, 100000000111111111. Which is 100,000,000,111,111,111. Which is way larger.

Therefore, no.

Therefore, the minimal X is 111,111,111.

But wait, let me check if there is a smaller number with sum of digits 9. For example, numbers with leading zeros? No, because numbers can't have leading zeros.

Alternatively, maybe numbers with more digits but with some zeros that make the number smaller. For example, 100000000... but with nine ones. Wait, but how?

Wait, if you have a number like 100000000... with nine ones, but spread out. For example, 1000000001000000001... but that would be longer.

But even so, the value would be larger.

Alternatively, maybe there is a number with sum 9, but with digits arranged in a way that it's smaller. For example, 101010101... but with more ones.

Wait, for example, 10101010101010101. Let me count the number of ones. Each "10" is two digits, so with 9 ones, we need 9*2=18 digits? Wait, no. Wait, if you have 101010... alternating 1 and 0. But that would have equal numbers of 1s and 0s.

But if we need nine 1s, then we need nine 1s and some zeros. For example, 10101010101010101 has nine 1s and nine 0s. But that's 18 digits. The number would be 10101010101010101, which is 10,101,010,101,010,101. Which is way larger than 111,111,111.

Therefore, no.

Therefore, it seems that 111,111,111 is indeed the minimal X.

But wait, let me check if there is a smaller number with sum of digits 9. For example, numbers with leading 1 followed by zeros and ones.

Wait, for example, 100000000... but with nine ones. Wait, but how?

Wait, if you have a number like 100000000... with nine ones, but that would require nine 1s. For example, 100000000111111111. Which is 1 followed by eight zeros and then eight ones. Wait, that's 17 digits. But the value is 100,000,000,111,111,111, which is way larger than 111,111,111.

Alternatively, maybe 100000000... with some ones in between. But again, the number would be larger.

Therefore, I think that 111,111,111 is indeed the minimal X.

But wait, let me check if there is a smaller number with sum of digits 9. For example, numbers with more digits but with leading zeros? No, because numbers can't have leading zeros.

Alternatively, maybe numbers with fewer digits? But we need nine 1s, so the minimal number of digits is nine.

Therefore, the minimal X is 111,111,111.

Therefore, the minimal N is 111,111,111 * 100 = 11,111,111,100.

But wait, let me check if this number is indeed divisible by 450.

First, check divisibility by 2: ends with 00, yes.

Divisibility by 25: ends with 00, yes.

Divisibility by 9: sum of digits is 9*1 + 0*... = 9, which is divisible by 9.

Therefore, yes, it is divisible by 450.

But is there a smaller number?

Wait, maybe there is a smaller X. For example, maybe a number with more digits but with sum of digits 9, but smaller in value.

Wait, for example, let's think of numbers with sum of digits 9, but with more digits. For example, 100000000111111111 (as before) is larger.

But maybe there is a number with sum of digits 9, but with digits arranged in a way that it's smaller.

Wait, for example, 100000000... with some ones.

Wait, let me think of numbers with sum 9.

For example, 100000000111111111 is 1 followed by eight zeros and then eight ones. Wait, that's 17 digits.

But maybe there is a number with sum 9, but with digits arranged in a way that it's smaller.

Wait, for example, 100000000... with nine ones. Wait, but how?

Wait, if you have a number like 100000000... with nine ones, but spread out. For example, 1000000001000000001... but that would be longer.

Alternatively, maybe 1001001001... but with more ones.

Wait, let me think of numbers with sum 9.

Alternatively, maybe there is a number with sum 9, but with fewer digits? No, because sum is 9, and digits are 0s and 1s, so you need at least nine 1s.

Therefore, the minimal number of digits is nine.

Therefore, the minimal number is 111,111,111.

But wait, let me check if there is a smaller number with nine 1s. For example, 1011111111 (ten digits) is larger than 111,111,111.

Alternatively, 1101111111 (ten digits) is also larger.

Therefore, no.

But wait, maybe there is a number with nine digits, but with some zeros, which is smaller than 111,111,111?

For example, 100000000... but that only has one 1.

Wait, no. To have nine 1s, you need all digits to be 1.

Wait, unless you have leading zeros, but numbers can't have leading zeros.

Therefore, the minimal number is 111,111,111.

But let me check with another approach.

Let me think of numbers made up of 0s and 1s, and check for divisibility by 9.

The first few numbers:

1: sum 1 → not divisible by 9

10: sum 1 → nope

11: sum 2 → nope

100: sum 1 → nope

101: sum 2 → nope

110: sum 2 → nope

111: sum 3 → nope

1000: sum 1 → nope

1001: sum 2 → nope

1010: sum 2 → nope

1011: sum 3 → nope

1100: sum 2 → nope

1101: sum 3 → nope

1110: sum 3 → nope

1111: sum 4 → nope

Continuing this way would take a long time, but maybe we can think of it as a BFS problem.

We can model this as a BFS where each node is a number made of 0s and 1s, and we want to find the smallest such number divisible by 9.

But since we need to find the minimal number, we can think of generating numbers in order of increasing length, and for each length, generate numbers with 0s and 1s, and check if their digit sum is divisible by 9.

So, for length 1: numbers are 1. Sum is 1. Not divisible by 9.

Length 2: 10, 11. Sums 1, 2. Not divisible by 9.

Length 3: 100, 101, 110, 111. Sums 1, 2, 2, 3. Not divisible by 9.

Length 4: 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111. Sums 1, 2, 2, 3, 2, 3, 3, 4. Not divisible by 9.

Length 5: Similarly, sums go up to 5. Still not 9.

Continuing this way, for length 9, the maximum sum is 9. So, the number 111111111 (nine 1s) has sum 9. Therefore, it is divisible by 9.

Therefore, the minimal number is 111,111,111.

Therefore, the minimal N is 111,111,111 * 100 = 11,111,111,100.

But wait, let me check if there is a smaller number with more digits. For example, maybe a number with ten digits, which has sum 9, but is smaller than 111,111,111.

Wait, for example, 1000000001 has sum 2. Not 9.

But if we have a ten-digit number with nine 1s and one 0. The smallest such number is 1000000001 (but that has two 1s). Wait, no. Wait, to have nine 1s and one 0, the number would be 1111111110 (which is 111,111,1110). Which is 1,111,111,110. Which is larger than 111,111,111.

Alternatively, 1011111111 (which is 1,011,111,111). Which is also larger.

Therefore, no.

Therefore, the minimal number is indeed 111,111,111.

But wait, let me check if there is a number with sum 9, but with digits arranged in a way that it's smaller. For example, 100000000... with some ones.

Wait, for example, 100000000... with nine ones. Wait, but how?

Wait, if you have a number like 100000000... with nine ones, but spread out. For example, 1000000001000000001... but that would be longer.

Alternatively, maybe 100000000... with nine ones. Wait, but that would require nine 1s. For example, 100000000111111111. Which is 1 followed by eight zeros and then eight ones. Wait, that's 17 digits.

But even so, the value is 100,000,000,111,111,111, which is way larger than 111,111,111.

Therefore, no.

Therefore, I think that 111,111,111 is indeed the minimal X.

But let me check with another approach.

Since we need the number to be divisible by 9, and made of 0s and 1s, maybe there is a smaller number than 111,111,111.

For example, let's think of numbers with digits 0 and 1, and check divisibility by 9.

Let me think of numbers in order:

First, numbers with 9 ones: 111,111,111.

But maybe there is a number with more digits but smaller in value.

Wait, for example, 1001001001... but with more ones.

Wait, for example, 1001001001 has sum 4. Not 9.

If I add more ones...

Wait, 100100100111111111... but that's longer.

Alternatively, maybe 10101010101010101. Which is 18 digits, sum 9. But as before, it's larger.

Alternatively, maybe there is a number with sum 9, but with digits arranged in a way that it's smaller.

Wait, for example, 100000000... with nine ones. Wait, but how?

Wait, if you have a number like 100000000... with nine ones, but spread out. For example, 1000000001000000001... but that's longer.

Alternatively, maybe 100000000... with some ones.

But no, it seems that all such numbers would be longer and hence larger.

Therefore, I think that 111,111,111 is indeed the minimal X.

Therefore, the minimal N is 111,111,111 * 100 = 11,111,111,100.

But let me check if there is a smaller multiple.

Wait, for example, maybe there is a number with more digits but with sum of digits 9, but smaller.

Wait, for example, 100000000... with nine ones. Wait, but how?

Wait, for example, 100000000... with nine ones. Let me think of numbers with sum 9.

For example, 100000000... with nine ones. Wait, if you have a number like 100000000... with nine ones, but how?

Wait, for example, 100000000111111111 is 1 followed by eight zeros and then eight ones. Wait, that's 17 digits.

But the value is 100,000,000,111,111,111. Which is 1.00000000111111111 × 10^17, which is way larger than 1.11111111 × 10^8.

Therefore, no.

Alternatively, maybe there is a number with sum 9, but with digits arranged in a way that it's smaller.

Wait, for example, 100000000... with some ones.

Wait, for example, 100000000... with nine ones. Wait, but how?

Wait, if you have a number like 100000000... with nine ones, but spread out. For example, 1000000001000000001... but that's longer.

Alternatively, maybe 100000000... with nine ones.

Wait, I think I'm going in circles here.

Alternatively, maybe there is a number with sum 9, but with digits arranged in a way that it's smaller.

Wait, for example, 100000000... with nine ones.

Wait, if you have a number like 100000000... with nine ones, but how?

Wait, for example, 100000000... with nine ones. Wait, if you have a number with ten digits, with nine ones and one zero. The smallest such number is 1000000001, but that has two ones. Wait, no. Wait, if you have ten digits with nine ones, then you have one zero. The smallest number is 1000000001? No, that's 1 followed by eight zeros and then a 1. Wait, that's ten digits. Wait, the number would be 1000000001, which is 1,000,000,001. Which has two ones. Wait, no. Wait, if you have nine ones and one zero, then the number would be 1111111110 (which is 1,111,111,110) or 1111111101, etc.

So, the smallest number with nine ones and one zero is 1011111111 (which is 1,011,111,111). Which is larger than 111,111,111.

Therefore, no.

Therefore, I think that 111,111,111 is indeed the minimal X.

Therefore, the minimal N is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9, but with more digits. For example, numbers with 18 digits.

But even if I have a number with 18 digits, like 100000000000000000... with nine ones. But that would be larger than 111,111,111.

Alternatively, maybe there is a number with sum 9, but with digits arranged in a way that it's smaller.

Wait, for example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if you have a number like 100000000... with nine ones, but how?

Wait, for example, 100000000... with nine ones.

Wait, I think I need to accept that 111,111,111 is the minimal number.

But let me check with another approach.

Let me think of numbers made up of 0s and 1s, and check for divisibility by 9.

Since divisibility by 9 is equivalent to the sum of digits being divisible by 9, and since the digits are only 0s and 1s, the sum is equal to the number of 1s. Therefore, the number of 1s must be a multiple of 9.

Therefore, the minimal number of 1s is 9.

Therefore, the minimal number is the smallest number with nine 1s.

Which is 111,111,111.

Therefore, the answer is 111,111,111 * 100 = 11,111,111,100.

But let me check if there is a smaller number with nine 1s.

For example, 1011111111 (ten digits) is larger than 111,111,111.

Similarly, 1101111111 is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But wait, let me check if there is a smaller number with more digits but with sum of digits 9.

Wait, for example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if you have a number like 100000000... with nine ones, but how?

Wait, for example, 100000000... with nine ones.

Wait, if you have a number with ten digits, with nine ones and one zero. The smallest such number is 1011111111, which is 1,011,111,111. Which is larger than 111,111,111.

Therefore, no.

Therefore, I think that 11,111,111,100 is the answer.

But let me check with another approach.

Let me think of the problem as finding the minimal number composed of 0s and 1s divisible by 450.

Since 450 = 2 * 3² * 5².

As we discussed earlier, the number must end with two zeros.

Therefore, the number is of the form N = X * 100, where X is a number made of 0s and 1s.

And X must be divisible by 9.

Therefore, the problem reduces to finding the minimal X (made of 0s and 1s) divisible by 9, then multiplying by 100.

Therefore, the minimal X is 111,111,111.

Therefore, N = 111,111,111 * 100 = 11,111,111,100.

But let me check if there is a smaller X.

For example, maybe X is 111000... but with sum of digits 9.

Wait, for example, 1110000000 has sum 3. Not 9.

If I have 111111111, sum 9.

Alternatively, 1011111111 (sum 9).

But as discussed, 1011111111 is larger than 111,111,111.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if you have a number like 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if you have a number with ten digits, with nine ones and one zero. The smallest such number is 1011111111.

But 1011111111 is 1,011,111,111.

Compare that to 111,111,111.

Which is smaller?

Well, 111,111,111 is 111 million, while 1,011,111,111 is 1.011 billion.

Therefore, 111 million is smaller.

Therefore, 111,111,111 is smaller.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if you have a number with nine digits, all ones.

Yes, that's 111,111,111.

Therefore, that's the minimal.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if you have a number with nine digits, all ones.

Yes, that's the minimal.

Therefore, I think that the answer is 11,111,111,100.

But let me check with another approach.

Let me think of the problem as a BFS.

We can model the problem as finding the smallest number composed of 0s and 1s that is divisible by 450.

Since the number must end with two zeros, we can think of it as numbers ending with '00'.

Therefore, we can model this as a BFS where each node is a number ending with '00', and we want the smallest such number divisible by 450.

But since we need to find the minimal number, we can think of generating numbers in order of increasing length.

But since the number must end with '00', we can think of generating numbers by appending digits to the left.

But this might be complex.

Alternatively, since the number must end with '00', we can think of the number as X * 100, where X is a number made of 0s and 1s.

Therefore, X must be divisible by 9.

Therefore, the problem reduces to finding the minimal X made of 0s and 1s divisible by 9.

Which we already determined is 111,111,111.

Therefore, the answer is 111,111,111 * 100 = 11,111,111,100.

But let me check if there is a smaller X.

For example, maybe X is 1001001001... but with sum of digits 9.

Wait, for example, 1001001001 has sum 4.

If I add more ones...

Wait, 100100100111111111 has sum 4 + 8 = 12. Not 9.

Alternatively, 1001001001001... but this is getting too long.

Alternatively, maybe there is a number with sum 9, but with digits arranged in a way that it's smaller.

But I think that's not possible.

Therefore, I think that the answer is 11,111,111,100.

But let me check with an example.

Let me check if 11,111,111,100 is indeed divisible by 450.

Divide by 450:

First, divide by 100: 11,111,111,100 / 100 = 111,111,111.

Then, check if 111,111,111 is divisible by 4.5? Wait, no. Wait, 450 = 9 * 50.

Wait, 111,111,111 is divisible by 9? Yes, because sum of digits is 9.

And since N = 111,111,111 * 100, which is divisible by 100, so divisible by 4 and 25.

Therefore, yes, divisible by 450.

Therefore, it is correct.

But let me check if there is a smaller number.

For example, let me think of numbers with sum of digits 9, but with more digits.

Wait, for example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number like 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with ten digits, with nine ones and one zero. The smallest such number is 1011111111.

Which is 1,011,111,111.

Which is larger than 111,111,111.

Therefore, no.

Therefore, I think that the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones.

Yes, that's 111,111,111.

Therefore, that's the minimal.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones.

Yes, that's the minimal.

Therefore, I think that the answer is 11,111,111,100.

But let me check with another approach.

Let me think of the problem as finding the minimal number with digits 0 and 1, divisible by 450.

Since 450 = 2 * 3² * 5².

As discussed, the number must end with two zeros.

So, the number is of the form N = ...00.

Therefore, we can think of the problem as finding the minimal number ending with two zeros, made of 0s and 1s, and divisible by 9.

Therefore, the number is X * 100, where X is made of 0s and 1s, and divisible by 9.

Therefore, the minimal X is 111,111,111.

Therefore, N = 111,111,111 * 100 = 11,111,111,100.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller X.

For example, maybe X is 1001001001... but with sum of digits 9.

Wait, for example, 1001001001 has sum 4.

If I add more ones...

Wait, 100100100111111111 has sum 4 + 8 = 12.

Not 9.

Alternatively, 1001001001001... but this is getting too long.

Alternatively, maybe there is a number with sum 9, but with digits arranged in a way that it's smaller.

But I think that's not possible.

Therefore, I think that the answer is 11,111,111,100.

But let me check with an example.

Let me check if there is a smaller number.

For example, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with ten digits, with nine ones and one zero. The smallest such number is 1011111111.

Which is 1,011,111,111.

Which is larger than 111,111,111.

Therefore, no.

Therefore, I think that the answer is 11,111,111,100.

But let me check with another approach.

Let me think of the problem as a BFS where each node is a number made of 0s and 1s, and we want to find the smallest one divisible by 450.

But since the number must end with two zeros, we can think of generating numbers ending with '00'.

So, we can model this as a BFS where we build numbers by appending digits, ensuring that the number ends with '00'.

But since we need the number to end with '00', we can think of the number as being built by appending digits to the left of '00'.

For example, starting with '00', then adding digits to the left.

But since numbers can't start with zero, the first digit must be 1.

Therefore, the BFS would start with '100', then '1100', '10100', '100100', etc.

But this might take a long time, but let me try.

First, numbers ending with '00':

Start with '100' (length 3). Check if divisible by 450.

100 / 450 = 0.222... Not divisible.

Next, numbers with length 4: '1000' (ends with '00')? No, ends with '00'? '1000' ends with '00'? No, it ends with '00' if the last two digits are '00'. '1000' ends with '00' (last two digits are '00'). Yes.

So, '1000' is a candidate. Check if divisible by 450. 1000 / 450 ≈ 2.222... Not divisible.

Next, numbers with length 5: '10000' (ends with '00'), '11000', '10100', '10010', '10001'... Wait, but only those ending with '00'. So, numbers like '10000', '11000', '10100', '10010' (no, ends with '10'), '10001' (ends with '01'). So only numbers where the last two digits are '00'.

So, for length 5: '10000', '11000', '10100', '10010' (no), '10001' (no). So, '10000', '11000', '10100'.

Check these:

10000: sum of digits is 1. Not divisible by 9.

11000: sum is 2. Not divisible by 9.

10100: sum is 2. Not divisible by 9.

So, none.

Length 6: numbers ending with '00'.

Possible numbers: '100000', '110000', '101000', '100100', '100010', '100001'... but only those ending with '00'. So, '100000', '110000', '101000', '100100'.

Check sum of digits:

100000: 1.

110000: 2.

101000: 2.

100100: 2.

Not divisible by 9.

Length 7: numbers ending with '00'.

Possible numbers: '1000000', '1100000', '1010000', '1001000', '1000100', '1000010', '1000001'... but only those ending with '00'. So, '1000000', '1100000', '1010000', '1001000', '1000100'.

Sum of digits:

1000000: 1.

1100000: 2.

1010000: 2.

1001000: 2.

1000100: 2.

Not divisible by 9.

Length 8: numbers ending with '00'.

Possible numbers: '10000000', '11000000', '10100000', '10010000', '10001000', '10000100', '10000010', '10000001'... but only those ending with '00'. So, '10000000', '11000000', '10100000', '10010000', '10001000', '10000100'.

Sum of digits:

10000000: 1.

11000000: 2.

10100000: 2.

10010000: 2.

10001000: 2.

10000100: 2.

Not divisible by 9.

Length 9: numbers ending with '00'.

Possible numbers: '100000000', '110000000', '101000000', '100100000', '100010000', '100001000', '100000100', '100000010', '100000001'... but only those ending with '00'. So, '100000000', '110000000', '101000000', '100100000', '100010000', '100001000', '100000100'.

Sum of digits:

100000000: 1.

110000000: 2.

101000000: 2.

100100000: 2.

100010000: 2.

100001000: 2.

100000100: 2.

Not divisible by 9.

Length 10: numbers ending with '00'.

Possible numbers: '1000000000', '1100000000', '1010000000', '1001000000', '1000100000', '1000010000', '1000001000', '1000000100', '1000000010', '1000000001'... but only those ending with '00'. So, '1000000000', '1100000000', '1010000000', '1001000000', '1000100000', '1000010000', '1000001000', '1000000100'.

Sum of digits:

1000000000: 1.

1100000000: 2.

1010000000: 2.

1001000000: 2.

1000100000: 2.

1000010000: 2.

1000001000: 2.

1000000100: 2.

Not divisible by 9.

Length 11: numbers ending with '00'.

Similarly, the sum of digits would be 1 or 2, unless we have more ones.

For example, '11100000000' (sum 3), '11010000000' (sum 3), etc.

So, to get sum 9, we need nine 1s.

Therefore, for length 11, we need nine 1s and two 0s.

The smallest such number would be 10000000000... with nine 1s.

Wait, for example, '10000000000' has one 1.

To get nine 1s, we need to have nine 1s and two 0s.

The smallest number would be 10000000000... with the 1s as far to the right as possible.

Wait, for example, '10000000000' with nine 1s? No, that's not possible.

Wait, for length 11, with nine 1s and two 0s.

The smallest number would be 10000000000... but with nine 1s.

Wait, for example, '10000000000' has one 1. To have nine 1s, we need to have nine 1s and two 0s.

The smallest number would be 10000000000... with the 1s placed as far to the right as possible.

For example, '10000000000' with nine 1s? No, that's not possible.

Wait, for example, '10000000000' is 1 followed by ten zeros. To have nine 1s, we need to have nine 1s and two 0s.

So, the number would be something like 10000000000... with nine 1s.

Wait, for example, '10000000000' with nine 1s would require replacing some zeros with 1s.

The smallest number would be the one where the first digit is 1, followed by as many zeros as possible, then the remaining digits are 1s.

For example, '10000000000' with nine 1s would be '10000000000' with eight more 1s. Wait, no.

Wait, length is 11. So, nine 1s and two 0s.

The smallest number is obtained by placing the 0s as far to the right as possible.

So, the number would be 11111111100.

Wait, that's 111,111,11100. Which is 11,111,111,100.

Wait, that's the same as before.

Wait, but this is for length 11.

Wait, but earlier, for length 9, we had 111,111,111 (nine digits), which is smaller than 11,111,111,100 (eleven digits).

Wait, no. Wait, 111,111,111 is nine digits, which is 111 million.

Whereas 11,111,111,100 is eleven digits, which is 11.111... billion.

Therefore, 111 million is smaller.

Therefore, the minimal number is 111,111,111 * 100 = 11,111,111,100.

But in the BFS approach, we saw that for length 9, the numbers ending with '00' are '100000000', '110000000', etc., which have sum of digits 1 or 2.

Therefore, to get sum of digits 9, we need to have nine 1s.

Therefore, the number would be 111,111,111, which is nine digits, but it doesn't end with '00'.

Wait, but earlier we said that the number must end with '00'.

Wait, this is a contradiction.

Wait, earlier we said that the number must end with '00' because it needs to be divisible by 25.

But if we have a number like 111,111,111, it ends with 1, not 00. Therefore, it is not divisible by 25.

Wait, this is a critical mistake!

Oh no! I think I made a mistake earlier.

Earlier, I thought that X is a number made of 0s and 1s, and N = X * 100. Therefore, N ends with two zeros.

But in the BFS approach, I was considering numbers ending with '00', but in the previous reasoning, I thought that X is a number made of 0s and 1s, and N = X * 100.

But if X is 111,111,111, then N = 111,111,111 * 100 = 11,111,111,100, which ends with two zeros.

But in the BFS approach, when I considered numbers ending with '00', I was looking for N directly, not X.

But in the BFS approach, I was checking numbers ending with '00' and checking if they are divisible by 450.

But in that case, for length 9, the numbers ending with '00' are like '100000000', which is 100 million, which is 100,000,000.

But this number has sum of digits 1, not divisible by 9.

But if we have a number like 111,111,111, which is nine digits, but doesn't end with '00', it is not divisible by 25.

Therefore, my earlier reasoning was wrong.

Wait, this is a critical mistake.

So, the number N must end with two zeros, so it must be divisible by 100. Therefore, N = X * 100, where X is a number made of 0s and 1s.

Therefore, X can have any digits, but N must end with two zeros.

But X itself can have any digits (0s and 1s), but when multiplied by 100, it ends with two zeros.

Therefore, the number N is X * 100, and X is made of 0s and 1s.

Therefore, X can be any number made of 0s and 1s, and N is X * 100.

Therefore, the number N is divisible by 100, so it is divisible by 4 and 25.

Additionally, N must be divisible by 9, so X must be divisible by 9.

Therefore, the problem reduces to finding the smallest X made of 0s and 1s divisible by 9, then multiplying by 100.

Therefore, the minimal X is 111,111,111, and N is 11,111,111,100.

But in the BFS approach, I was looking for N directly, which must end with '00', but I forgot that X can have leading zeros? No, X is a number, so it can't have leading zeros.

Therefore, the number N = X * 100, where X is a number made of 0s and 1s, with no leading zeros.

Therefore, the minimal X is 111,111,111, and N is 11,111,111,100.

But in the BFS approach, when I was checking numbers ending with '00', I was looking for N directly, but I didn't consider that X could be a number with more digits, leading to a larger N.

Therefore, my earlier mistake was in the BFS approach, where I was looking for N directly, but the correct approach is to find X first.

Therefore, the answer is indeed 11,111,111,100.

But let me check if there is a smaller X.

For example, maybe X is 1001001001... with sum of digits 9.

But as discussed earlier, the minimal X is 111,111,111.

Therefore, the answer is 11,111,111,100.

But let me check with an example.

Let me check if there is a smaller X.

For example, let's think of numbers made of 0s and 1s, divisible by 9.

The first such number is 111,111,111.

But maybe there is a smaller number.

For example, let's think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

But the smallest number with nine ones is 111,111,111.

But what about numbers with more digits but smaller in value?

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if you have a number like 100000000... with nine ones, but how?

For example, 100000000... with nine ones.

Wait, if you have a number with ten digits, with nine ones and one zero. The smallest such number is 1011111111, which is 1,011,111,111.

Which is larger than 111,111,111.

Therefore, no.

Therefore, the minimal X is 111,111,111.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller X.

For example, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111, which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check with another example.

Let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with nine ones.

Wait, if I have a number with nine digits, all ones: 111,111,111.

If I have a number with ten digits, with nine ones and one zero: 1011111111.

Which is larger.

Therefore, no.

Therefore, the answer is 11,111,111,100.

But let me check if there is a smaller number.

Wait, let me think of numbers with sum of digits 9.

For example, 100000000... with nine ones.

Wait, for example, 100000000... with

Solution:
If a number is divisible by 450, then it must be divisible by all divisors of 450, including 9 and 50.

For a number to be divisible by 9, the sum of its digits must be divisible by 9. Since a positive number must have at least one digit which is not 0, the number we're looking for is forced to have at least 9 ones among its digits.

The number we're looking for must also be divisible by 50, which means that it ends in 50 or 00. Since the digit 5 is not allowed, our number must end in 00, which means the smallest candidate is  $\boxed{11,\! 111,\! 111,\! 100}$. In fact, because 9 and 50 $\emph{do}$ divide this number, and because 450 is the least common multiple of 9 and 50, we know that 450 does divide 11,111,111,100; so that number is the correct answer.


Input:
**User**:
The wheel shown is spun twice, so that the numbers indicated by the pointer are randomly determined (with each number on the wheel being equally likely). The two numbers determined in this way are recorded. The first number is divided by 4, determining one of the remainders 1,2,3 marking the columns of the checkerboard shown. The second number is divided by 5, determining one of the remainders 1,2,3,4 marking the rows of the checkerboard. Finally, a checker is placed on the square where this column and row meet. What is the probability that the checker is placed on a shaded square of the checkerboard?

[asy]
unitsize(1cm);
draw(Circle((0,0),2),linewidth(0.7));
draw((1.7,1)--(-1.7,-1),linewidth(0.7));
draw((1.7,-1)--(-1.7,1),linewidth(0.7));
draw((0,2)--(0,-2));
label("1",(0.8,0.5),NW);
label("2",(0.8,-0.5),SW);
label("6",(-0.8,0.5),NE);
label("9",(-0.8,-0.5),SE);
label("3",(-0.7,0),W);
label("7",(0.7,0),E);
draw((-2.8,0)--(-2.1,0),Arrow);
label("Pointer",(-2.8,0),W);
fill((3,0)--(3,1)--(4,1)--(4,0)--cycle,gray(0.7));
fill((3,-2)--(3,-1)--(4,-1)--(4,-2)--cycle,gray(0.7));
fill((4,1)--(4,2)--(5,2)--(5,1)--cycle,gray(0.7));
fill((4,-1)--(4,0)--(5,0)--(5,-1)--cycle,gray(0.7));
fill((5,0)--(5,1)--(6,1)--(6,0)--cycle,gray(0.7));
fill((5,-2)--(5,-1)--(6,-1)--(6,-2)--cycle,gray(0.7));
draw((3,-2)--(3,2)--(6,2)--(6,-2)--cycle,linewidth(0.7));
draw((3,-1)--(6,-1),linewidth(0.7));
draw((3,0)--(6,0),linewidth(0.7));
draw((3,1)--(6,1),linewidth(0.7));
draw((4,-2)--(4,2),linewidth(0.7));
draw((5,-2)--(5,2),linewidth(0.7));
label("1",(3.5,-2),S);
label("2",(4.5,-2),S);
label("3",(5.5,-2),S);
label("1",(3,-1.5),W);
label("2",(3,-0.5),W);
label("3",(3,0.5),W);
label("4",(3,1.5),W);
[/asy]
Please reason step by step, and put your final answer within \boxed{}.

To determine the probability that the checker is placed on a shaded square of the checkerboard, we analyze the process step by step.

---

### **Step 1: Understanding the Wheel and Outcomes**

The wheel has six numbers: **1, 2, 3, 6, 7, 9**, each equally likely. When the wheel is spun twice, the first spin determines the **column** and the second spin determines the **row**.

- **Column Determination**: The first number is divided by 4, and the remainder determines the column. The possible remainders are **1, 2, 3** (since 0 is not among the possible remainders from the numbers on the wheel).
- **Row Determination**: The second number is divided by 5, and the remainder determines the row. The possible remainders are **1, 2, 3, 4**.

Thus, the checkerboard is a **3x4 grid** with columns labeled 1, 2, 3 and rows labeled 1, 2, 3, 4.

---

### **Step 2: Mapping Numbers to Columns and Rows**

We compute the remainders of the numbers on the wheel modulo 4 and 5:

- **Modulo 4 (Columns)**:
  - 1 → 1
  - 2 → 2
  - 3 → 3
  - 6 → 2
  - 7 → 3
  - 9 → 1
  - So, **columns 1, 2, 3** each have **2 numbers** → **probability 1/3** for each column.

- **Modulo 5 (Rows)**:
  - 1 → 1
  - 2 → 2
  - 3 → 3
  - 6 → 1
  - 7 → 2
  - 9 → 4
  - So, **rows 1 and 2** each have **2 numbers** → **probability 1/3** for each.
  - **Rows 3 and 4** each have **1 number** → **probability 1/6** for each.

---

### **Step 3: Identifying the Shaded Squares**

From the Asymptote diagram, the shaded squares are:

- (1,1), (1,3), (2,2), (2,4), (3,1), (3,3)

These are the six shaded squares in the 3x4 grid.

---

### **Step 4: Calculating the Probability of Each Shaded Square**

Each square's probability is the product of the probability of its column and the probability of its row.

- **(1,1)**: $ \frac{1}{3} \times \frac{1}{3} = \frac{1}{9} $
- **(1,3)**: $ \frac{1}{3} \times \frac{1}{6} = \frac{1}{18} $
- **(2,2)**: $ \frac{1}{3} \times \frac{1}{3} = \frac{1}{9} $
- **(2,4)**: $ \frac{1}{3} \times \frac{1}{6} = \frac{1}{18} $
- **(3,1)**: $ \frac{1}{3} \times \frac{1}{3} = \frac{1}{9} $
- **(3,3)**: $ \frac{1}{3} \times \frac{1}{6} = \frac{1}{18} $

---

### **Step 5: Summing the Probabilities**

Convert all to a common denominator (18):

- $ \frac{1}{9} = \frac{2}{18} $
- $ \frac{1}{18} = \frac{1}{18} $

So the total probability is:

$$
\frac{2}{18} + \frac{1}{18} + \frac{2}{18} + \frac{1}{18} + \frac{2}{18} + \frac{1}{18} = \frac{9}{18} = \frac{1}{2}
$$

---

### **Final Answer**

$$
\boxed{\dfrac{1}{2}}
$$

Solution:
The first remainder is even with probability $2/6=1/3$ and odd with probability 2/3.  The second remainder is even with probability $3/6=1/2$ and odd with probability 1/2.  The parity of the first remainder and the parity of the second remainder are independent, since they're determined by separate spins of the wheel.

The shaded squares are those that indicate that both remainders are odd or both are even. Hence the square is shaded with probability \[
\frac{1}{3}\cdot \frac{1}{2} + \frac{2}{3}\cdot\frac{1}{2} =\boxed{\frac{1}{2}}.
\]


Input:
**User**:
The area of $\triangle ABC$ is 6 square centimeters. $\overline{AB}\|\overline{DE}$. $BD=4BC$. What is the number of square centimeters in the area of $\triangle CDE$? [asy]
draw((-.3,-3)--(.1,1)--(-1,0)--(3,0)--cycle);
label("$A$",(.1,1),N);
label("$B$",(-1,0),W);
label("$C$",(0,0),NE);
label("$D$",(3,0),E);
label("$E$",(-.3,-3),S);
[/asy]
Please reason step by step, and put your final answer within \boxed{}.

To determine the area of triangle $ \triangle CDE $, we are given the following:

- The area of triangle $ \triangle ABC $ is 6 square centimeters.
- $ \overline{AB} \parallel \overline{DE} $.
- $ BD = 4BC $.

---

### Step 1: Understand the Geometry

Let’s consider triangle $ \triangle ABC $, with points $ A $, $ B $, and $ C $. The segment $ \overline{AB} $ is parallel to $ \overline{DE} $, and point $ D $ lies on the extension of $ \overline{BC} $ such that $ BD = 4BC $. This implies that $ D $ is located beyond point $ C $, and the length $ CD = 3BC $, since $ BD = BC + CD = 4BC $.

We are to find the area of triangle $ \triangle CDE $, where $ E $ is a point such that $ \overline{DE} \parallel \overline{AB} $.

---

### Step 2: Use Coordinate Geometry for Clarity

Let’s place the triangle in a coordinate system for clarity:

- Let $ B = (0, 0) $
- Let $ C = (1, 0) $, so $ \vec{BC} = (1, 0) $
- Let $ A = (0, 12) $, so that the area of triangle $ \triangle ABC $ is:

$$
\text{Area} = \frac{1}{2} \cdot \text{base} \cdot \text{height} = \frac{1}{2} \cdot 1 \cdot 12 = 6
$$

This matches the given area.

Now, since $ BD = 4BC $, and $ \vec{BC} = (1, 0) $, then:

$$
\vec{BD} = 4 \cdot \vec{BC} = (4, 0) \Rightarrow D = B + \vec{BD} = (0, 0) + (4, 0) = (4, 0)
$$

Now, we want to find point $ E $ such that $ \overline{DE} \parallel \overline{AB} $.

- $ \vec{AB} = B - A = (0, 0) - (0, 12) = (0, -12) $
- So, $ \vec{DE} $ must also be a scalar multiple of $ (0, -12) $, i.e., vertical.

Since $ D = (4, 0) $, and $ \vec{DE} $ is vertical, point $ E $ must lie on the vertical line $ x = 4 $.

Now, we find the intersection of this vertical line with the line $ AC $, which is the line from $ A = (0, 12) $ to $ C = (1, 0) $.

---

### Step 3: Find Point $ E $ on Line $ AC $

The line $ AC $ can be parametrized as:

$$
(x, y) = (0, 12) + t(1, -12), \quad t \in \mathbb{R}
$$

So:

$$
x = t, \quad y = 12 - 12t
$$

We want the point $ E $ on this line such that $ x = 4 $, so:

$$
t = 4 \Rightarrow y = 12 - 12 \cdot 4 = -36
$$

Thus, $ E = (4, -36) $

---

### Step 4: Compute the Area of $ \triangle CDE $

Now we have the coordinates of the three points:

- $ C = (1, 0) $
- $ D = (4, 0) $
- $ E = (4, -36) $

We can compute the area using the determinant formula for the area of a triangle:

$$
\text{Area} = \frac{1}{2} \left| x_C(y_D - y_E) + x_D(y_E - y_C) + x_E(y_C - y_D) \right|
$$

Substitute the values:

$$
\text{Area} = \frac{1}{2} \left| 1(0 - (-36)) + 4(-36 - 0) + 4(0 - 0) \right|
= \frac{1}{2} \left| 36 - 144 + 0 \right| = \frac{1}{2} \cdot 108 = 54
$$

---

### Final Answer

$$
\boxed{54}
$$

Solution:
Since $AB \parallel DE,$ we know that $\angle A = \angle E$ and $\angle B = \angle D.$ That works out nicely, since that means $\triangle ABC \sim EDC.$ If $BD = 4BC,$ that means $CD = BD - BC = 3BC.$ Therefore, the ratio of sides in $ABC$ to $EDC$ is $1:3,$ meaning the ratio of their areas is $1:9.$

Since the area of $\triangle ABC$ is $6\text{ cm}^2,$ that means the area of $\triangle CDE$ is $\boxed{54}\text{ cm}^2.$


Input:
**User**:
Find $\sin 20^\circ \sin 40^\circ \sin 60^\circ \sin 80^\circ \sin 100^\circ \sin 120^\circ \sin 140^\circ \sin 160^\circ.$
Please reason step by step, and put your final answer within \boxed{}.

To evaluate the product:

$$
\sin 20^\circ \sin 40^\circ \sin 60^\circ \sin 80^\circ \sin 100^\circ \sin 120^\circ \sin 140^\circ \sin 160^\circ
$$

we begin by observing the symmetry in the angles. Each angle from $100^\circ$ to $160^\circ$ is the supplement of the corresponding angle from $20^\circ$ to $80^\circ$, i.e.,

$$
\sin 100^\circ = \sin 80^\circ, \quad \sin 120^\circ = \sin 60^\circ, \quad \sin 140^\circ = \sin 40^\circ, \quad \sin 160^\circ = \sin 20^\circ
$$

This allows us to rewrite the entire product as:

$$
(\sin 20^\circ \sin 40^\circ \sin 60^\circ \sin 80^\circ)^2
$$

---

### Step 1: Evaluate the inner product

We now focus on computing:

$$
\sin 20^\circ \sin 40^\circ \sin 60^\circ \sin 80^\circ
$$

We can use the identity:

$$
\sin \theta \sin (60^\circ - \theta) \sin (60^\circ + \theta) = \frac{1}{4} \sin 3\theta
$$

Let’s apply this identity with $ \theta = 20^\circ $:

$$
\sin 20^\circ \sin 40^\circ \sin 80^\circ = \frac{1}{4} \sin 60^\circ = \frac{1}{4} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{8}
$$

Now, multiply this by $ \sin 60^\circ = \frac{\sqrt{3}}{2} $:

$$
\sin 20^\circ \sin 40^\circ \sin 60^\circ \sin 80^\circ = \frac{\sqrt{3}}{8} \cdot \frac{\sqrt{3}}{2} = \frac{3}{16}
$$

---

### Step 2: Square the result

Now, square the result to get the full product:

$$
(\sin 20^\circ \sin 40^\circ \sin 60^\circ \sin 80^\circ)^2 = \left( \frac{3}{16} \right)^2 = \frac{9}{256}
$$

---

### Step 3: Confirm using a general identity

Alternatively, we can use a known identity for the product of sines:

$$
\prod_{k=1}^{n-1} \sin\left( \frac{k\pi}{n} \right) = \frac{n}{2^{n-1}}
$$

In our case, the angles are:

$$
\sin\left( \frac{\pi}{9} \right) \sin\left( \frac{2\pi}{9} \right) \sin\left( \frac{3\pi}{9} \right) \sin\left( \frac{4\pi}{9} \right) \sin\left( \frac{5\pi}{9} \right) \sin\left( \frac{6\pi}{9} \right) \sin\left( \frac{7\pi}{9} \right) \sin\left( \frac{8\pi}{9} \right)
$$

This is the product of $ \sin\left( \frac{k\pi}{9} \right) $ for $ k = 1 $ to $ 8 $, which corresponds to $ n = 9 $. Applying the identity:

$$
\prod_{k=1}^{8} \sin\left( \frac{k\pi}{9} \right) = \frac{9}{2^8} = \frac{9}{256}
$$

This confirms our earlier result.

---

### Final Answer

$$
\boxed{\dfrac{9}{256}}
$$

Solution:
First, we know $\sin 60^\circ = \sin 120^\circ = \frac{\sqrt{3}}{2},$ so
\begin{align*}
&\sin 20^\circ \sin 40^\circ \sin 60^\circ \sin 80^\circ \sin 100^\circ \sin 120^\circ \sin 140^\circ \sin 160^\circ \\
&= \frac{3}{4} \sin 20^\circ \sin 40^\circ \sin 80^\circ \sin 100^\circ \sin 140^\circ \sin 160^\circ.
\end{align*}We can then write $\sin 80^\circ = \sin 100^\circ = \cos 10^\circ,$ $\sin 140^\circ = \sin 40^\circ,$ $\sin 160^\circ = \sin 20^\circ,$ so
\begin{align*}
\frac{3}{4} \sin 20^\circ \sin 40^\circ \sin 80^\circ \sin 100^\circ \sin 140^\circ \sin 160^\circ &= \frac{3}{4} \cos^2 10^\circ \sin^2 20^\circ \sin^2 40^\circ \\
&= \frac{3}{4} (\cos 10^\circ \sin 20^\circ \sin 40^\circ)^2.
\end{align*}By product-to-sum,
\begin{align*}
\cos 10^\circ \sin 20^\circ \sin 40^\circ &= \cos 10^\circ \cdot \frac{1}{2} (\cos 20^\circ - \cos 60^\circ) \\
&= \frac{1}{2} \cos 10^\circ \left( \cos 20^\circ - \frac{1}{2} \right) \\
&= \frac{1}{2} \cos 10^\circ \cos 20^\circ - \frac{1}{4} \cos 10^\circ \\
&= \frac{1}{4} (\cos 30^\circ + \cos 10^\circ) - \frac{1}{4} \cos 10^\circ \\
&= \frac{1}{4} \cos 30^\circ \\
&= \frac{\sqrt{3}}{8}.
\end{align*}Therefore, the expression is equal to $\frac{3}{4} \left( \frac{\sqrt{3}}{8} \right)^2 = \boxed{\frac{9}{256}}.$