Explanation:
Okay, so I need to find an integer n between 0 and 17 (since 0 ≤ n < 18) such that n is congruent to -11213141 modulo 18. Hmm, negative numbers in modular arithmetic can be a bit tricky, so I need to remember how to handle that. Let me think...

First, maybe I can convert the negative number into a positive one by adding multiples of 18 until it's within the range of 0 to 17. But before I do that, maybe there's a smarter way. I remember that for modulus operations, we can take the number modulo 18 directly, but since the number is so large, -11213141, it's going to be tedious to compute directly. Maybe I can find the remainder when dividing by 18 by breaking down the number?

Wait, there's a property that allows us to find the remainder modulo 18 by using the fact that 18 is 2*9, so maybe I can use modulo 2 and modulo 9, then combine them? But maybe that's overcomplicating. Alternatively, since 18 is a small number, perhaps I can compute the value of -11213141 modulo 18 by first finding 11213141 modulo 18 and then taking the negative of that?

Yes, that might work. Let me recall that if I can find 11213141 mod 18, then -11213141 mod 18 would be equal to (- (11213141 mod 18)) mod 18. That is, if I first find the positive remainder of 11213141 divided by 18, and then subtract that from 18 to get a positive equivalent.

But how do I compute 11213141 mod 18? Well, there's another trick for modulus with 9, because 10 ≡ 1 mod 9, so the sum of the digits of the number modulo 9 is equal to the number modulo 9. Similarly, for modulus 2, it's just the last digit. But modulus 18 is a bit more complex. Wait, but maybe I can use the fact that 18 = 2*9 and use the Chinese Remainder Theorem?

Let me try that. Let me find 11213141 modulo 2 and modulo 9, then combine them.

Starting with modulo 2: The number is 11213141. The last digit is 1, which is odd, so 11213141 ≡ 1 mod 2.

Now modulo 9: Sum the digits of 11213141. Let me do that step by step:

1 + 1 + 2 + 1 + 3 + 1 + 4 + 1.

Calculating that:

1 + 1 = 2; 2 + 2 = 4; 4 + 1 = 5; 5 + 3 = 8; 8 + 1 = 9; 9 + 4 = 13; 13 + 1 = 14.

So the sum is 14. Then, 14 mod 9 is 14 - 9 = 5. Therefore, 11213141 ≡ 5 mod 9.

So, the number is congruent to 1 mod 2 and 5 mod 9. Now, we need to find a number n such that n ≡ 1 mod 2 and n ≡ 5 mod 9, and then n would be congruent to 11213141 mod 18. Wait, but actually, since we need the original number mod 18, then once we have the value mod 18, we can take its negative and then adjust accordingly. Let me check.

But maybe there's a better way. Alternatively, since I need -11213141 mod 18, which is equal to (- (11213141 mod 18)) mod 18. So if I can compute 11213141 mod 18, then subtract that from 18 to get the positive equivalent?

Alternatively, since modulus is distributive over addition and subtraction, perhaps I can find -11213141 mod 18 by breaking down the number into parts.

Alternatively, maybe there's a way to compute 11213141 mod 18 by using the fact that 10 ≡ 10 mod 18, 10^2 ≡ 100 mod 18. Let me compute 100 mod 18. 18*5=90, so 100-90=10, so 10^2 ≡ 10 mod 18. Then 10^3 = 10^2 * 10 ≡ 10*10 = 100 ≡ 10 mod 18. Wait, this seems like 10^k mod 18 cycles?

Wait, let's check:

10^1 mod 18 = 10

10^2 = 100 mod 18. 18*5 = 90, 100 - 90 = 10, so 10^2 ≡ 10 mod 18

Similarly, 10^3 = 10^2 * 10 ≡ 10*10 = 100 ≡ 10 mod 18

So 10^k ≡ 10 mod 18 for k ≥ 1. Interesting. Therefore, any power of 10 beyond 10^1 is congruent to 10 mod 18. Wait, is that correct?

Wait, that seems odd. Let me check with 10^1=10, 10^2=100≡10 mod 18, 10^3=1000. Let me compute 10^3 mod 18. 1000 divided by 18: 18*55=990, so 1000-990=10. So yes, 10^3≡10 mod 18. Similarly, 10^4=10^3*10≡10*10=100≡10 mod 18. So indeed, for any exponent k ≥ 1, 10^k ≡ 10 mod 18. That seems to be a pattern.

So, given that, perhaps I can write the number 11213141 as a sum of digits multiplied by powers of 10. Let me write it out:

The number 11213141 can be broken down as:

1*10^7 + 1*10^6 + 2*10^5 + 1*10^4 + 3*10^3 + 1*10^2 + 4*10^1 + 1*10^0.

Now, since for exponents k ≥ 1, 10^k ≡ 10 mod 18. So for each term:

10^7 ≡ 10 mod 18,

10^6 ≡ 10 mod 18,

Similarly, 10^5 ≡ 10 mod 18,

10^4 ≡ 10 mod 18,

10^3 ≡ 10 mod 18,

10^2 ≡ 10 mod 18,

10^1 ≡ 10 mod 18,

and 10^0 ≡ 1 mod 18.

Therefore, substituting:

1*10 + 1*10 + 2*10 + 1*10 + 3*10 + 1*10 + 4*10 + 1*1.

Let me compute each term:

First, the coefficients for the powers of 10 from 10^7 down to 10^1:

1*10 = 10,

1*10 = 10,

2*10 = 20,

1*10 = 10,

3*10 = 30,

1*10 = 10,

4*10 = 40,

and then the last term is 1*1=1.

Adding all those up:

Let me add them step by step.

Start with 10 + 10 = 20,

20 + 20 = 40,

40 + 10 = 50,

50 + 30 = 80,

80 + 10 = 90,

90 + 40 = 130,

130 + 1 = 131.

So the total is 131. Therefore, 11213141 ≡ 131 mod 18. Now, I need to compute 131 mod 18. Let me divide 131 by 18.

18*7=126, 18*7=126, 131-126=5. So 131 ≡ 5 mod 18. Therefore, 11213141 ≡ 5 mod 18. Therefore, -11213141 ≡ -5 mod 18. Now, to express this as a positive number between 0 and 17, we can add 18 to -5: -5 + 18 = 13. Therefore, -11213141 ≡ 13 mod 18. Therefore, the integer n is 13.

Wait, but let me check my calculations again to make sure I didn't make any mistakes, because it's easy to mess up with so many steps.

First, breaking down the number into digits:

Number is 11213141. Let me write down each digit:

Digits from left to right: 1, 1, 2, 1, 3, 1, 4, 1. So positions from 10^7 down to 10^0.

Yes, that's correct. Then, for exponents ≥1, 10^k ≡10 mod18. Therefore, each coefficient multiplied by 10, except the last term which is 1.

So the sum is:

(1 + 1 + 2 + 1 + 3 + 1 + 4)*10 + 1.

Calculating coefficients: 1+1=2; 2+2=4; 4+1=5; 5+3=8; 8+1=9; 9+4=13. So total is 13*10 +1 = 130 +1=131. Then 131 divided by 18: 18*7=126, remainder 5. So yes, 131 mod18=5. Therefore, -5 mod18 is 13. Therefore, n=13.

Alternatively, maybe I can check using another method. Let me try to compute 11213141 mod 18 by another way, just to verify.

Another approach is to use the fact that 18 divides 9*2, so maybe compute the number mod 18 by using the fact that 10 ≡ 10 mod 18. Alternatively, using the technique of taking the alternating sum or something else?

Wait, maybe another approach is to compute the number step by step, reducing modulo 18 at each step. Let me try that.

Starting with 0, then process each digit from left to right.

But let me think. The number is 11213141.

Let me write it as 1,1,2,1,3,1,4,1.

Starting with 0. Multiply by 10 and add next digit:

First digit is 1:

current = 0 * 10 + 1 = 1 mod 18.

Second digit is 1:

current = 1 * 10 + 1 = 11 mod 18.

Third digit is 2:

current = 11 * 10 + 2 = 110 + 2 = 112 mod 18.

Compute 112 mod 18: 18*6=108, 112-108=4. So current is 4 mod18.

Fourth digit is 1:

current = 4 * 10 + 1 = 40 +1 = 41 mod 18. 18*2=36, so 41-36=5. Current is 5 mod18.

Fifth digit is 3:

current = 5 * 10 + 3 = 50 +3=53 mod18. 18*2=36, 18*3=54. So 53 - 54 = -1, but mod18 that's 17. Wait, 53 divided by 18 is 2*18=36, 53-36=17. So current is 17 mod18.

Sixth digit is 1:

current = 17 *10 +1 = 170 +1=171 mod18. Let me divide 171 by 18: 18*9=162. 171-162=9. So current is 9 mod18.

Seventh digit is 4:

current = 9 *10 +4=90 +4=94 mod18. Divide 94 by 18: 18*5=90, 94-90=4. So current is 4 mod18.

Eighth digit is 1:

current = 4*10 +1=40 +1=41 mod18. Which we already saw is 5 mod18. Wait, but earlier when I processed the digits step by step, after the sixth digit, current was 9, then seventh digit 4 gives 4, and eighth digit 1 gives 5? Wait, but earlier when I broke down the number as sum of digits with coefficients, I got 131 mod18=5. So here, processing digit by digit gives me 5 mod18. So that's the same as before. So then, 11213141 ≡5 mod18, so -11213141 ≡ -5 mod18≡13 mod18. Therefore, n=13.

Therefore, that seems consistent with my first method. So that gives me more confidence that n=13 is the correct answer.

But just to be absolutely sure, maybe I can check with another method. Let me compute 11213141 divided by 18 and find the remainder.

Alternatively, since 18=2*9, and we already found that 11213141≡1 mod2 and 5 mod9. Then, since n ≡1 mod2 and n≡5 mod9. Let me find such n between 0-17.

So looking for numbers between 0-17 that are 5 mod9. The numbers are 5, 14. Because 5+9=14, which is still less than 18. Then check which of these is ≡1 mod2.

5 mod2 is 1, since 5 is odd. 14 mod2 is 0, since even. Therefore, the number is 5 mod9 and 1 mod2 is 5. Therefore, 11213141 ≡5 mod18. Therefore, -11213141≡-5 mod18≡13 mod18. Therefore, n=13.

So that's the same answer as before. Therefore, seems like n=13 is correct.

Alternatively, let's check with actual division. Let me try to compute 11213141 divided by 18.

But that might be tedious, but perhaps I can compute 11213141 divided by 18. Let me note that 18*600,000 = 10,800,000.

Subtract that from 11,213,141: 11,213,141 - 10,800,000 = 413,141.

Then, 18*20,000 = 360,000. Subtract that from 413,141: 413,141 - 360,000 = 53,141.

Then, 18*3,000 = 54,000. That's larger than 53,141. Try 18*2,900 = 18*(2000 + 900) = 36,000 + 16,200 = 52,200. Subtract that from 53,141: 53,141 - 52,200 = 941.

Now, 18*50 = 900. Subtract that from 941: 941 - 900 = 41.

Now, 18*2 = 36. Subtract that: 41 - 36 = 5.

So total quotient is 600,000 + 20,000 + 2,900 + 50 + 2 = 600,000 + 20,000 = 620,000; 620,000 + 2,900 = 622,900; 622,900 + 50 = 622,950; 622,950 + 2 = 622,952. The remainder is 5. Therefore, 11213141 divided by 18 gives quotient 622,952 and remainder 5. Therefore, 11213141 ≡5 mod18. Therefore, -11213141 ≡-5 mod18≡13 mod18. So n=13.

This confirms it again. So regardless of the method, the remainder is 5, so the negative of that is -5 ≡13 mod18. So n=13.

Therefore, I think 13 is the correct answer.

But just to make sure, let me check if 13 is indeed congruent to -11213141 modulo 18.

Let me compute 13 + 11213141, and check if it's divisible by 18.

13 + 11213141 = 11213154. Let's check if 11213154 is divisible by 18. Since 18=2*9, check divisibility by 2 and 9.

Divisible by 2: The last digit is 4, which is even, so yes.

Divisible by 9: Sum the digits of 11213154. Let's compute:

1 + 1 + 2 + 1 + 3 + 1 + 5 + 4.

Calculating step by step: 1+1=2, 2+2=4, 4+1=5, 5+3=8, 8+1=9, 9+5=14, 14+4=18. The sum is 18, which is divisible by 9. Therefore, 11213154 is divisible by both 2 and 9, hence divisible by 18. Therefore, 13 ≡ -11213141 mod18. Perfect, that confirms it.

Therefore, the integer n is 13.

**Final Answer**
The integer $ n $ is \boxed{13}.

To solve the problem of finding an integer $ n $ such that:

$$
0 \leq n < 18 \quad \text{and} \quad n \equiv -11213141 \pmod{18}
$$

we aim to determine the value of $ n $ that is congruent to $ -11213141 \mod 18 $, within the specified range.

---

### Step 1: Understand What the Question Is Asking

We are given a large negative number and asked to find a small (positive) integer $ n $ such that:

$$
n \equiv -11213141 \pmod{18}
$$

This means $ n $ should be a representative of the same congruence class as $ -11213141 $ modulo 18, but in the range $ 0 \leq n < 18 $.

---

### Step 2: Convert Negative Modulo to Positive

Since $ -11213141 $ is negative, we can convert it to a positive residue modulo 18 by computing:

$$
-11213141 \mod 18 \equiv (- (11213141 \mod 18)) \mod 18
$$

So, the key is to first compute $ 11213141 \mod 18 $.

---

### Step 3: Compute $ 11213141 \mod 18 $

Let’s compute this using a modular breakdown technique that is effective for large numbers:

#### Use the property: $ 10^k \equiv 10 \mod 18 $ for $ k \geq 1 $

We can write the number $ 11213141 $ as a linear combination of powers of 10:

$$
11213141 = 1 \cdot 10^7 + 1 \cdot 10^6 + 2 \cdot 10^5 + 1 \cdot 10^4 + 3 \cdot 10^3 + 1 \cdot 10^2 + 4 \cdot 10^1 + 1 \cdot 10^0
$$

Since $ 10^k \equiv 10 \mod 18 $ for all $ k \geq 1 $, and $ 10^0 \equiv 1 \mod 18 $, we rewrite the expression modulo 18:

$$
\begin{align*}
1 \cdot 10 + 1 \cdot 10 + 2 \cdot 10 + 1 \cdot 10 + 3 \cdot 10 + 1 \cdot 10 + 4 \cdot 10 + 1 \cdot 1 \\
&= (1+1+2+1+3+1+4) \cdot 10 + 1 \\
&= 13 \cdot 10 + 1 = 130 + 1 = 131 \\
\end{align*}
$$

Now compute $ 131 \mod 18 $:

$$
131 \div 18 = 7 \text{ remainder } 5 \Rightarrow 131 \equiv 5 \mod 18
$$

Thus:

$$
11213141 \equiv 5 \mod 18
$$

---

### Step 4: Compute $ -11213141 \mod 18 $

Since $ -11213141 \equiv -5 \mod 18 $, we now find the positive equivalent of $ -5 $ modulo 18:

$$
-5 + 18 = 13
$$

Therefore,

$$
-11213141 \equiv 13 \mod 18
$$

---

### Step 5: Final Answer

The value of $ n $ that satisfies:

$$
0 \leq n < 18 \quad \text{and} \quad n \equiv -11213141 \pmod{18}
$$

is:

$$
\boxed{13}
$$

[Assistant]
Explanation:
Okay, so I need to find an integer n between 0 and 17 (since 0 ≤ n < 18) such that n is congruent to -11213141 modulo 18. Hmm, negative numbers in modular arithmetic can be a bit tricky, so I need to remember how to handle that. Let me think...

First, maybe I can convert the negative number into a positive one by adding multiples of 18 until it's within the range of 0 to 17. But before I do that, maybe there's a smarter way. I remember that for modulus operations, we can take the number modulo 18 directly, but since the number is so large, -11213141, it's going to be tedious to compute directly. Maybe I can find the remainder when dividing by 18 by breaking down the number?

Wait, there's a property that allows us to find the remainder modulo 18 by using the fact that 18 is 2*9, so maybe I can use modulo 2 and modulo 9, then combine them? But maybe that's overcomplicating. Alternatively, since 18 is a small number, perhaps I can compute the value of -11213141 modulo 18 by first finding 11213141 modulo 18 and then taking the negative of that?

Yes, that might work. Let me recall that if I can find 11213141 mod 18, then -11213141 mod 18 would be equal to (- (11213141 mod 18)) mod 18. That is, if I first find the positive remainder of 11213141 divided by 18, and then subtract that from 18 to get a positive equivalent.

But how do I compute 11213141 mod 18? Well, there's another trick for modulus with 9, because 10 ≡ 1 mod 9, so the sum of the digits of the number modulo 9 is equal to the number modulo 9. Similarly, for modulus 2, it's just the last digit. But modulus 18 is a bit more complex. Wait, but maybe I can use the fact that 18 = 2*9 and use the Chinese Remainder Theorem?

Let me try that. Let me find 11213141 modulo 2 and modulo 9, then combine them.

Starting with modulo 2: The number is 11213141. The last digit is 1, which is odd, so 11213141 ≡ 1 mod 2.

Now modulo 9: Sum the digits of 11213141. Let me do that step by step:

1 + 1 + 2 + 1 + 3 + 1 + 4 + 1.

Calculating that:

1 + 1 = 2; 2 + 2 = 4; 4 + 1 = 5; 5 + 3 = 8; 8 + 1 = 9; 9 + 4 = 13; 13 + 1 = 14.

So the sum is 14. Then, 14 mod 9 is 14 - 9 = 5. Therefore, 11213141 ≡ 5 mod 9.

So, the number is congruent to 1 mod 2 and 5 mod 9. Now, we need to find a number n such that n ≡ 1 mod 2 and n ≡ 5 mod 9, and then n would be congruent to 11213141 mod 18. Wait, but actually, since we need the original number mod 18, then once we have the value mod 18, we can take its negative and then adjust accordingly. Let me check.

But maybe there's a better way. Alternatively, since I need -11213141 mod 18, which is equal to (- (11213141 mod 18)) mod 18. So if I can compute 11213141 mod 18, then subtract that from 18 to get the positive equivalent?

Alternatively, since modulus is distributive over addition and subtraction, perhaps I can find -11213141 mod 18 by breaking down the number into parts.

Alternatively, maybe there's a way to compute 11213141 mod 18 by using the fact that 10 ≡ 10 mod 18, 10^2 ≡ 100 mod 18. Let me compute 100 mod 18. 18*5=90, so 100-90=10, so 10^2 ≡ 10 mod 18. Then 10^3 = 10^2 * 10 ≡ 10*10 = 100 ≡ 10 mod 18. Wait, this seems like 10^k mod 18 cycles?

Wait, let's check:

10^1 mod 18 = 10

10^2 = 100 mod 18. 18*5 = 90, 100 - 90 = 10, so 10^2 ≡ 10 mod 18

Similarly, 10^3 = 10^2 * 10 ≡ 10*10 = 100 ≡ 10 mod 18

So 10^k ≡ 10 mod 18 for k ≥ 1. Interesting. Therefore, any power of 10 beyond 10^1 is congruent to 10 mod 18. Wait, is that correct?

Wait, that seems odd. Let me check with 10^1=10, 10^2=100≡10 mod 18, 10^3=1000. Let me compute 10^3 mod 18. 1000 divided by 18: 18*55=990, so 1000-990=10. So yes, 10^3≡10 mod 18. Similarly, 10^4=10^3*10≡10*10=100≡10 mod 18. So indeed, for any exponent k ≥ 1, 10^k ≡ 10 mod 18. That seems to be a pattern.

So, given that, perhaps I can write the number 11213141 as a sum of digits multiplied by powers of 10. Let me write it out:

The number 11213141 can be broken down as:

1*10^7 + 1*10^6 + 2*10^5 + 1*10^4 + 3*10^3 + 1*10^2 + 4*10^1 + 1*10^0.

Now, since for exponents k ≥ 1, 10^k ≡ 10 mod 18. So for each term:

10^7 ≡ 10 mod 18,

10^6 ≡ 10 mod 18,

Similarly, 10^5 ≡ 10 mod 18,

10^4 ≡ 10 mod 18,

10^3 ≡ 10 mod 18,

10^2 ≡ 10 mod 18,

10^1 ≡ 10 mod 18,

and 10^0 ≡ 1 mod 18.

Therefore, substituting:

1*10 + 1*10 + 2*10 + 1*10 + 3*10 + 1*10 + 4*10 + 1*1.

Let me compute each term:

First, the coefficients for the powers of 10 from 10^7 down to 10^1:

1*10 = 10,

1*10 = 10,

2*10 = 20,

1*10 = 10,

3*10 = 30,

1*10 = 10,

4*10 = 40,

and then the last term is 1*1=1.

Adding all those up:

Let me add them step by step.

Start with 10 + 10 = 20,

20 + 20 = 40,

40 + 10 = 50,

50 + 30 = 80,

80 + 10 = 90,

90 + 40 = 130,

130 + 1 = 131.

So the total is 131. Therefore, 11213141 ≡ 131 mod 18. Now, I need to compute 131 mod 18. Let me divide 131 by 18.

18*7=126, 18*7=126, 131-126=5. So 131 ≡ 5 mod 18. Therefore, 11213141 ≡ 5 mod 18. Therefore, -11213141 ≡ -5 mod 18. Now, to express this as a positive number between 0 and 17, we can add 18 to -5: -5 + 18 = 13. Therefore, -11213141 ≡ 13 mod 18. Therefore, the integer n is 13.

Wait, but let me check my calculations again to make sure I didn't make any mistakes, because it's easy to mess up with so many steps.

First, breaking down the number into digits:

Number is 11213141. Let me write down each digit:

Digits from left to right: 1, 1, 2, 1, 3, 1, 4, 1. So positions from 10^7 down to 10^0.

Yes, that's correct. Then, for exponents ≥1, 10^k ≡10 mod18. Therefore, each coefficient multiplied by 10, except the last term which is 1.

So the sum is:

(1 + 1 + 2 + 1 + 3 + 1 + 4)*10 + 1.

Calculating coefficients: 1+1=2; 2+2=4; 4+1=5; 5+3=8; 8+1=9; 9+4=13. So total is 13*10 +1 = 130 +1=131. Then 131 divided by 18: 18*7=126, remainder 5. So yes, 131 mod18=5. Therefore, -5 mod18 is 13. Therefore, n=13.

Alternatively, maybe I can check using another method. Let me try to compute 11213141 mod 18 by another way, just to verify.

Another approach is to use the fact that 18 divides 9*2, so maybe compute the number mod 18 by using the fact that 10 ≡ 10 mod 18. Alternatively, using the technique of taking the alternating sum or something else?

Wait, maybe another approach is to compute the number step by step, reducing modulo 18 at each step. Let me try that.

Starting with 0, then process each digit from left to right.

But let me think. The number is 11213141.

Let me write it as 1,1,2,1,3,1,4,1.

Starting with 0. Multiply by 10 and add next digit:

First digit is 1:

current = 0 * 10 + 1 = 1 mod 18.

Second digit is 1:

current = 1 * 10 + 1 = 11 mod 18.

Third digit is 2:

current = 11 * 10 + 2 = 110 + 2 = 112 mod 18.

Compute 112 mod 18: 18*6=108, 112-108=4. So current is 4 mod18.

Fourth digit is 1:

current = 4 * 10 + 1 = 40 +1 = 41 mod 18. 18*2=36, so 41-36=5. Current is 5 mod18.

Fifth digit is 3:

current = 5 * 10 + 3 = 50 +3=53 mod18. 18*2=36, 18*3=54. So 53 - 54 = -1, but mod18 that's 17. Wait, 53 divided by 18 is 2*18=36, 53-36=17. So current is 17 mod18.

Sixth digit is 1:

current = 17 *10 +1 = 170 +1=171 mod18. Let me divide 171 by 18: 18*9=162. 171-162=9. So current is 9 mod18.

Seventh digit is 4:

current = 9 *10 +4=90 +4=94 mod18. Divide 94 by 18: 18*5=90, 94-90=4. So current is 4 mod18.

Eighth digit is 1:

current = 4*10 +1=40 +1=41 mod18. Which we already saw is 5 mod18. Wait, but earlier when I processed the digits step by step, after the sixth digit, current was 9, then seventh digit 4 gives 4, and eighth digit 1 gives 5? Wait, but earlier when I broke down the number as sum of digits with coefficients, I got 131 mod18=5. So here, processing digit by digit gives me 5 mod18. So that's the same as before. So then, 11213141 ≡5 mod18, so -11213141 ≡ -5 mod18≡13 mod18. Therefore, n=13.

Therefore, that seems consistent with my first method. So that gives me more confidence that n=13 is the correct answer.

But just to be absolutely sure, maybe I can check with another method. Let me compute 11213141 divided by 18 and find the remainder.

Alternatively, since 18=2*9, and we already found that 11213141≡1 mod2 and 5 mod9. Then, since n ≡1 mod2 and n≡5 mod9. Let me find such n between 0-17.

So looking for numbers between 0-17 that are 5 mod9. The numbers are 5, 14. Because 5+9=14, which is still less than 18. Then check which of these is ≡1 mod2.

5 mod2 is 1, since 5 is odd. 14 mod2 is 0, since even. Therefore, the number is 5 mod9 and 1 mod2 is 5. Therefore, 11213141 ≡5 mod18. Therefore, -11213141≡-5 mod18≡13 mod18. Therefore, n=13.

So that's the same answer as before. Therefore, seems like n=13 is correct.

Alternatively, let's check with actual division. Let me try to compute 11213141 divided by 18.

But that might be tedious, but perhaps I can compute 11213141 divided by 18. Let me note that 18*600,000 = 10,800,000.

Subtract that from 11,213,141: 11,213,141 - 10,800,000 = 413,141.

Then, 18*20,000 = 360,000. Subtract that from 413,141: 413,141 - 360,000 = 53,141.

Then, 18*3,000 = 54,000. That's larger than 53,141. Try 18*2,900 = 18*(2000 + 900) = 36,000 + 16,200 = 52,200. Subtract that from 53,141: 53,141 - 52,200 = 941.

Now, 18*50 = 900. Subtract that from 941: 941 - 900 = 41.

Now, 18*2 = 36. Subtract that: 41 - 36 = 5.

So total quotient is 600,000 + 20,000 + 2,900 + 50 + 2 = 600,000 + 20,000 = 620,000; 620,000 + 2,900 = 622,900; 622,900 + 50 = 622,950; 622,950 + 2 = 622,952. The remainder is 5. Therefore, 11213141 divided by 18 gives quotient 622,952 and remainder 5. Therefore, 11213141 ≡5 mod18. Therefore, -11213141 ≡-5 mod18≡13 mod18. So n=13.

This confirms it again. So regardless of the method, the remainder is 5, so the negative of that is -5 ≡13 mod18. So n=13.

Therefore, I think 13 is the correct answer.

But just to make sure, let me check if 13 is indeed congruent to -11213141 modulo 18.

Let me compute 13 + 11213141, and check if it's divisible by 18.

13 + 11213141 = 11213154. Let's check if 11213154 is divisible by 18. Since 18=2*9, check divisibility by 2 and 9.

Divisible by 2: The last digit is 4, which is even, so yes.

Divisible by 9: Sum the digits of 11213154. Let's compute:

1 + 1 + 2 + 1 + 3 + 1 + 5 + 4.

Calculating step by step: 1+1=2, 2+2=4, 4+1=5, 5+3=8, 8+1=9, 9+5=14, 14+4=18. The sum is 18, which is divisible by 9. Therefore, 11213154 is divisible by both 2 and 9, hence divisible by 18. Therefore, 13 ≡ -11213141 mod18. Perfect, that confirms it.

Therefore, the integer n is 13.

**Final Answer**
The integer $ n $ is \boxed{13}.

To solve the problem of finding an integer $ n $ such that:

$$
0 \leq n < 18 \quad \text{and} \quad n \equiv -11213141 \pmod{18}
$$

we aim to determine the value of $ n $ that is congruent to $ -11213141 \mod 18 $, within the specified range.

---

### Step 1: Understand What the Question Is Asking

We are given a large negative number and asked to find a small (positive) integer $ n $ such that:

$$
n \equiv -11213141 \pmod{18}
$$

This means $ n $ should be a representative of the same congruence class as $ -11213141 $ modulo 18, but in the range $ 0 \leq n < 18 $.

---

### Step 2: Convert Negative Modulo to Positive

Since $ -11213141 $ is negative, we can convert it to a positive residue modulo 18 by computing:

$$
-11213141 \mod 18 \equiv (- (11213141 \mod 18)) \mod 18
$$

So, the key is to first compute $ 11213141 \mod 18 $.

---

### Step 3: Compute $ 11213141 \mod 18 $

Let’s compute this using a modular breakdown technique that is effective for large numbers:

#### Use the property: $ 10^k \equiv 10 \mod 18 $ for $ k \geq 1 $

We can write the number $ 11213141 $ as a linear combination of powers of 10:

$$
11213141 = 1 \cdot 10^7 + 1 \cdot 10^6 + 2 \cdot 10^5 + 1 \cdot 10^4 + 3 \cdot 10^3 + 1 \cdot 10^2 + 4 \cdot 10^1 + 1 \cdot 10^0
$$

Since $ 10^k \equiv 10 \mod 18 $ for all $ k \geq 1 $, and $ 10^0 \equiv 1 \mod 18 $, we rewrite the expression modulo 18:

$$
\begin{align*}
1 \cdot 10 + 1 \cdot 10 + 2 \cdot 10 + 1 \cdot 10 + 3 \cdot 10 + 1 \cdot 10 + 4 \cdot 10 + 1 \cdot 1 \\
&= (1+1+2+1+3+1+4) \cdot 10 + 1 \\
&= 13 \cdot 10 + 1 = 130 + 1 = 131 \\
\end{align*}
$$

Now compute $ 131 \mod 18 $:

$$
131 \div 18 = 7 \text{ remainder } 5 \Rightarrow 131 \equiv 5 \mod 18
$$

Thus:

$$
11213141 \equiv 5 \mod 18
$$

---

### Step 4: Compute $ -11213141 \mod 18 $

Since $ -11213141 \equiv -5 \mod 18 $, we now find the positive equivalent of $ -5 $ modulo 18:

$$
-5 + 18 = 13
$$

Therefore,

$$
-11213141 \equiv 13 \mod 18
$$

---

### Step 5: Final Answer

The value of $ n $ that satisfies:

$$
0 \leq n < 18 \quad \text{and} \quad n \equiv -11213141 \pmod{18}
$$

is:

$$
\boxed{13}
$$

Solution: An integer is divisible by $18$ if and only if the sum of its digits is divisible by $9$ and the last digit is even (meaning it is divisible by both 9 and 2).  The sum of the digits of $-11213141$ is 14.  Since $-11213141$ is negative, this number is 5 $\textit{less than}$ a multiple of 9.  The number is 4 $\textit{more than}$ a multiple of 9.  Subtracting 4 gives  \[-11213141 = -11213145+4.\]Since $-11213145$ has a digit sum of 18, this number is a multiple of 9.  However this is not a multiple of 18 so we need to subtract 9 again: \[-11213141 = -11213154+13.\]Now the number $-11213154$ is a multiple of 18, so the answer is $\boxed{13}$. $$-11213141\equiv 13\pmod {18}.$$


Explanation:
Okay, let's see. I need to find the value of f(3) given that f(-3) = 2 for the function f(x) = ax⁴ - bx² + x + 5. Hmm. Alright, let's start by writing down what's given.

First, the function is a quartic polynomial: f(x) = ax⁴ - bx² + x + 5. The coefficients are a and b, which are constants that I don't know yet. But I do know that when x is -3, the value of the function is 2. So, f(-3) = 2. My goal is to find f(3).

Let me write out f(-3) using the given function:

f(-3) = a(-3)^4 - b(-3)^2 + (-3) + 5 = 2.

Let me compute each term step by step.

First term: (-3)^4. Well, (-3)^4 is (-3)*(-3)*(-3)*(-3). Let me compute that. (-3)*(-3) is 9, then 9*(-3) is -27, and -27*(-3) is 81. So the first term is 81a.

Second term: -b(-3)^2. Let me compute (-3)^2 first. That's 9. Then multiply by -b: -b*9, which is -9b.

Third term: + (-3). That's straightforward, it's -3.

Fourth term: +5. That's just 5.

Putting it all together:

81a - 9b - 3 + 5 = 2.

Simplify the constants: -3 + 5 is 2. So the equation becomes:

81a - 9b + 2 = 2.

Subtract 2 from both sides:

81a - 9b = 0.

Hmm, so 81a - 9b = 0. Let me write that as an equation:

81a - 9b = 0. Maybe I can simplify this equation by dividing both sides by 9. Let me try that:

(81a)/9 - (9b)/9 = 0/9 => 9a - b = 0.

So, 9a - b = 0, which means that b = 9a.

Okay, so that's a relationship between a and b. But I need to find f(3). Let me write out f(3) using the same function:

f(3) = a(3)^4 - b(3)^2 + 3 + 5.

Let me compute each term:

First term: 3^4 is 81, so 81a.

Second term: -b*(3)^2. 3 squared is 9, so -9b.

Third term: +3.

Fourth term: +5.

So f(3) = 81a - 9b + 3 + 5. Simplify constants: 3 + 5 is 8. So f(3) = 81a - 9b + 8.

Wait a second! From earlier, we found that 81a - 9b = 0. Therefore, substituting into f(3):

f(3) = 0 + 8 = 8?

Wait, so is it just 8? That seems straightforward. Let me check my steps again to be sure.

Starting with f(-3) = 2:

f(-3) = 81a - 9b - 3 + 5 = 81a - 9b + 2 = 2. Therefore 81a -9b = 0. Correct.

Then f(3) = 81a -9b + 8. Since 81a -9b is 0, then f(3) is 0 + 8 = 8. So 8?

But let me verify that there isn't a mistake here. Let me recompute f(-3) and f(3).

Original function: f(x) = ax⁴ - bx² + x + 5.

f(-3):

(-3)^4 is 81, so term 1 is 81a.

(-3)^2 is 9, so term 2 is -b*9 = -9b.

Then the linear term is x, so when x = -3, term 3 is -3.

Then the constant term is +5. So total: 81a -9b -3 +5 = 81a -9b +2. Which equals 2. Therefore 81a -9b = 0. Correct.

Similarly, f(3):

(3)^4 is 81a, same as before.

(3)^2 is 9, so term 2 is -b*9 = -9b.

Linear term is x = 3, so +3.

Constant term +5. So total: 81a -9b +3 +5 = 81a -9b +8. And since 81a -9b is 0, then f(3)= 8.

So the answer is 8? That seems too straightforward. Wait, but maybe there is something else?

Wait, the function is f(x) = ax⁴ - bx² + x + 5. Let me check if there is any parity or symmetry here. Let me think about even and odd functions.

The function is a combination of even and odd terms. Let's see:

ax⁴ is an even function, since x⁴ is even.

- bx² is also an even function.

x is an odd function.

Then +5 is constant, even function.

So overall, f(x) can be written as even terms plus odd terms. So f(x) = (ax⁴ - bx² + 5) + x.

Therefore, if we consider f(-x):

f(-x) = a(-x)^4 - b(-x)^2 + (-x) + 5 = ax⁴ - bx² - x + 5.

Compare that with f(x):

f(x) = ax⁴ - bx² + x + 5.

So, f(-x) = [ax⁴ - bx² + 5] - x. Whereas f(x) = [ax⁴ - bx² +5] + x.

Therefore, if we denote g(x) = ax⁴ - bx² +5, which is even, then f(x) = g(x) + x, and f(-x) = g(x) - x.

Therefore, f(-x) + f(x) = [g(x) - x] + [g(x) + x] = 2g(x), which is even.

Similarly, f(-x) - f(x) = -2x, which is odd.

But maybe there is another way to approach this. Since we have f(-x) and f(x), perhaps we can relate them. Let me try that.

Let me compute f(-x) + f(x). From above:

f(-x) + f(x) = 2g(x) = 2(ax⁴ - bx² +5).

But in our case, we can compute f(-3) and f(3) and relate them.

Let me do that. Let me denote f(3) as something. Let me write:

f(-3) + f(3) = 2(ax⁴ - bx² +5) evaluated at x=3. Wait, but actually, more directly:

From above, f(-x) + f(x) = 2*(ax⁴ - bx² + 5). So for x=3:

f(-3) + f(3) = 2*(a*(3)^4 - b*(3)^2 + 5)

Let me compute that:

a*81 - b*9 +5 multiplied by 2.

But from earlier, we found that 81a - 9b = 0. Therefore, 81a - 9b +5 = 0 +5 =5. Therefore, f(-3) + f(3) = 2*5 = 10.

But we know that f(-3) = 2, so substituting:

2 + f(3) = 10 => f(3) = 10 -2 = 8.

Same result! So that confirms it. Therefore, regardless of the values of a and b, as long as they satisfy the condition f(-3)=2, f(3) must be 8.

Wait, that seems to be the case. So even though we don't know the exact values of a and b, the relationship between f(-3) and f(3) allows us to compute f(3) directly.

Therefore, the answer is 8.

But let me think again if there's any chance I could have messed up the signs or the terms. For example, when I calculated f(-3), the linear term was -3, right?

Yes, because f(-3) = a*(-3)^4 - b*(-3)^2 + (-3) + 5. So that term is + (-3). So that's -3. Then +5, so -3 +5 is +2. Then the rest is 81a -9b. Then 81a -9b +2 =2, so 81a -9b=0.

And when I calculated f(3), it was 81a -9b + 3 +5, which is 81a -9b +8. Since 81a -9b is 0, that gives 8.

Alternatively, using the symmetric relation, f(-x) + f(x) = 2*(ax⁴ - bx² +5). So when x=3, f(-3) + f(3) = 2*(ax⁴ - bx² +5). But ax⁴ - bx² is 81a -9b. Which is 0, so that becomes 2*(0 +5)=10. Therefore, 2 + f(3)=10 => f(3)=8.

Yes, that's consistent.

Therefore, I think the answer is 8.

But just to be thorough, let me suppose some values for a and b that satisfy 9a - b =0, and check if f(3)=8.

For example, let me choose a=1, then b=9a=9. So let me compute f(3):

f(3)= a*81 - b*9 +3 +5. Plugging in a=1, b=9:

81*1 -9*9 +3 +5.

Calculate:

81 - 81 + 3 +5 = 0 + 8 = 8. Perfect.

Another example: let me take a=2, so b=18.

f(3)=81*2 -9*18 +3 +5.

Calculate: 162 - 162 + 3 +5 = 0 +8=8. Same result.

Another example, take a=0. Then b=0.

f(3)= 0 -0 +3 +5=8.

Wait, that's even simpler. If a=0 and b=0, then f(x)=0x⁴ -0x² +x +5 =x +5. Then f(-3)= -3 +5=2, which matches the given condition. Then f(3)=3 +5=8.

So that works as well.

Therefore, regardless of the values of a and b (as long as they satisfy the condition from f(-3)=2), f(3) is always 8.

Therefore, the answer is 8.

**Final Answer**
The value of $ f(3) $ is \boxed{8}.

To find the value of $ f(3) $, we're given the function:

$$
f(x) = ax^4 - bx^2 + x + 5
$$

and also the value of $ f(-3) = 2 $.

---

### Step 1: Use the given function to express $ f(-3) $

Plug $ x = -3 $ into the function:

$$
f(-3) = a(-3)^4 - b(-3)^2 + (-3) + 5
$$

Now compute each term:

- $ (-3)^4 = 81 $
- $ (-3)^2 = 9 $

So:

$$
f(-3) = 81a - 9b - 3 + 5 = 81a - 9b + 2
$$

We are told that $ f(-3) = 2 $, so:

$$
81a - 9b + 2 = 2
$$

Subtract 2 from both sides:

$$
81a - 9b = 0
$$

This gives a simple relationship between $ a $ and $ b $:

$$
81a - 9b = 0 \quad \Rightarrow \quad 9a - b = 0 \quad \Rightarrow \quad b = 9a
$$

---

### Step 2: Compute $ f(3) $

Now evaluate $ f(3) $ using the same function:

$$
f(3) = a(3)^4 - b(3)^2 + 3 + 5 = 81a - 9b + 3 + 5
$$

Simplify the constants:

$$
f(3) = 81a - 9b + 8
$$

From Step 1, we found that $ 81a - 9b = 0 $, so:

$$
f(3) = 0 + 8 = 8
$$

---

### Step 3: Verification

To ensure correctness, we observe the function has even and odd components:

- $ ax^4 - bx^2 $: even
- $ x $: odd
- $ 5 $: even

Let’s define:

$$
g(x) = ax^4 - bx^2 + 5 \quad \text{(even function)}
$$
$$
f(x) = g(x) + x
$$
$$
f(-x) = g(x) - x
$$

Then:

$$
f(x) + f(-x) = 2g(x)
$$

So for any $ x $, including $ x = 3 $:

$$
f(3) + f(-3) = 2g(3)
$$

We are told $ f(-3) = 2 $, and we are solving for $ f(3) $. So:

$$
f(3) + 2 = 2g(3)
$$

Now compute $ g(3) $:

$$
g(3) = a(3)^4 - b(3)^2 + 5 = 81a - 9b + 5 = 0 + 5 = 5
$$

So:

$$
f(3) + 2 = 2 \cdot 5 = 10 \quad \Rightarrow \quad f(3) = 8
$$

---

### Final Answer

$$
\boxed{8}
$$

[Assistant]
Explanation:
Okay, let's see. I need to find the value of f(3) given that f(-3) = 2 for the function f(x) = ax⁴ - bx² + x + 5. Hmm. Alright, let's start by writing down what's given.

First, the function is a quartic polynomial: f(x) = ax⁴ - bx² + x + 5. The coefficients are a and b, which are constants that I don't know yet. But I do know that when x is -3, the value of the function is 2. So, f(-3) = 2. My goal is to find f(3).

Let me write out f(-3) using the given function:

f(-3) = a(-3)^4 - b(-3)^2 + (-3) + 5 = 2.

Let me compute each term step by step.

First term: (-3)^4. Well, (-3)^4 is (-3)*(-3)*(-3)*(-3). Let me compute that. (-3)*(-3) is 9, then 9*(-3) is -27, and -27*(-3) is 81. So the first term is 81a.

Second term: -b(-3)^2. Let me compute (-3)^2 first. That's 9. Then multiply by -b: -b*9, which is -9b.

Third term: + (-3). That's straightforward, it's -3.

Fourth term: +5. That's just 5.

Putting it all together:

81a - 9b - 3 + 5 = 2.

Simplify the constants: -3 + 5 is 2. So the equation becomes:

81a - 9b + 2 = 2.

Subtract 2 from both sides:

81a - 9b = 0.

Hmm, so 81a - 9b = 0. Let me write that as an equation:

81a - 9b = 0. Maybe I can simplify this equation by dividing both sides by 9. Let me try that:

(81a)/9 - (9b)/9 = 0/9 => 9a - b = 0.

So, 9a - b = 0, which means that b = 9a.

Okay, so that's a relationship between a and b. But I need to find f(3). Let me write out f(3) using the same function:

f(3) = a(3)^4 - b(3)^2 + 3 + 5.

Let me compute each term:

First term: 3^4 is 81, so 81a.

Second term: -b*(3)^2. 3 squared is 9, so -9b.

Third term: +3.

Fourth term: +5.

So f(3) = 81a - 9b + 3 + 5. Simplify constants: 3 + 5 is 8. So f(3) = 81a - 9b + 8.

Wait a second! From earlier, we found that 81a - 9b = 0. Therefore, substituting into f(3):

f(3) = 0 + 8 = 8?

Wait, so is it just 8? That seems straightforward. Let me check my steps again to be sure.

Starting with f(-3) = 2:

f(-3) = 81a - 9b - 3 + 5 = 81a - 9b + 2 = 2. Therefore 81a -9b = 0. Correct.

Then f(3) = 81a -9b + 8. Since 81a -9b is 0, then f(3) is 0 + 8 = 8. So 8?

But let me verify that there isn't a mistake here. Let me recompute f(-3) and f(3).

Original function: f(x) = ax⁴ - bx² + x + 5.

f(-3):

(-3)^4 is 81, so term 1 is 81a.

(-3)^2 is 9, so term 2 is -b*9 = -9b.

Then the linear term is x, so when x = -3, term 3 is -3.

Then the constant term is +5. So total: 81a -9b -3 +5 = 81a -9b +2. Which equals 2. Therefore 81a -9b = 0. Correct.

Similarly, f(3):

(3)^4 is 81a, same as before.

(3)^2 is 9, so term 2 is -b*9 = -9b.

Linear term is x = 3, so +3.

Constant term +5. So total: 81a -9b +3 +5 = 81a -9b +8. And since 81a -9b is 0, then f(3)= 8.

So the answer is 8? That seems too straightforward. Wait, but maybe there is something else?

Wait, the function is f(x) = ax⁴ - bx² + x + 5. Let me check if there is any parity or symmetry here. Let me think about even and odd functions.

The function is a combination of even and odd terms. Let's see:

ax⁴ is an even function, since x⁴ is even.

- bx² is also an even function.

x is an odd function.

Then +5 is constant, even function.

So overall, f(x) can be written as even terms plus odd terms. So f(x) = (ax⁴ - bx² + 5) + x.

Therefore, if we consider f(-x):

f(-x) = a(-x)^4 - b(-x)^2 + (-x) + 5 = ax⁴ - bx² - x + 5.

Compare that with f(x):

f(x) = ax⁴ - bx² + x + 5.

So, f(-x) = [ax⁴ - bx² + 5] - x. Whereas f(x) = [ax⁴ - bx² +5] + x.

Therefore, if we denote g(x) = ax⁴ - bx² +5, which is even, then f(x) = g(x) + x, and f(-x) = g(x) - x.

Therefore, f(-x) + f(x) = [g(x) - x] + [g(x) + x] = 2g(x), which is even.

Similarly, f(-x) - f(x) = -2x, which is odd.

But maybe there is another way to approach this. Since we have f(-x) and f(x), perhaps we can relate them. Let me try that.

Let me compute f(-x) + f(x). From above:

f(-x) + f(x) = 2g(x) = 2(ax⁴ - bx² +5).

But in our case, we can compute f(-3) and f(3) and relate them.

Let me do that. Let me denote f(3) as something. Let me write:

f(-3) + f(3) = 2(ax⁴ - bx² +5) evaluated at x=3. Wait, but actually, more directly:

From above, f(-x) + f(x) = 2*(ax⁴ - bx² + 5). So for x=3:

f(-3) + f(3) = 2*(a*(3)^4 - b*(3)^2 + 5)

Let me compute that:

a*81 - b*9 +5 multiplied by 2.

But from earlier, we found that 81a - 9b = 0. Therefore, 81a - 9b +5 = 0 +5 =5. Therefore, f(-3) + f(3) = 2*5 = 10.

But we know that f(-3) = 2, so substituting:

2 + f(3) = 10 => f(3) = 10 -2 = 8.

Same result! So that confirms it. Therefore, regardless of the values of a and b, as long as they satisfy the condition f(-3)=2, f(3) must be 8.

Wait, that seems to be the case. So even though we don't know the exact values of a and b, the relationship between f(-3) and f(3) allows us to compute f(3) directly.

Therefore, the answer is 8.

But let me think again if there's any chance I could have messed up the signs or the terms. For example, when I calculated f(-3), the linear term was -3, right?

Yes, because f(-3) = a*(-3)^4 - b*(-3)^2 + (-3) + 5. So that term is + (-3). So that's -3. Then +5, so -3 +5 is +2. Then the rest is 81a -9b. Then 81a -9b +2 =2, so 81a -9b=0.

And when I calculated f(3), it was 81a -9b + 3 +5, which is 81a -9b +8. Since 81a -9b is 0, that gives 8.

Alternatively, using the symmetric relation, f(-x) + f(x) = 2*(ax⁴ - bx² +5). So when x=3, f(-3) + f(3) = 2*(ax⁴ - bx² +5). But ax⁴ - bx² is 81a -9b. Which is 0, so that becomes 2*(0 +5)=10. Therefore, 2 + f(3)=10 => f(3)=8.

Yes, that's consistent.

Therefore, I think the answer is 8.

But just to be thorough, let me suppose some values for a and b that satisfy 9a - b =0, and check if f(3)=8.

For example, let me choose a=1, then b=9a=9. So let me compute f(3):

f(3)= a*81 - b*9 +3 +5. Plugging in a=1, b=9:

81*1 -9*9 +3 +5.

Calculate:

81 - 81 + 3 +5 = 0 + 8 = 8. Perfect.

Another example: let me take a=2, so b=18.

f(3)=81*2 -9*18 +3 +5.

Calculate: 162 - 162 + 3 +5 = 0 +8=8. Same result.

Another example, take a=0. Then b=0.

f(3)= 0 -0 +3 +5=8.

Wait, that's even simpler. If a=0 and b=0, then f(x)=0x⁴ -0x² +x +5 =x +5. Then f(-3)= -3 +5=2, which matches the given condition. Then f(3)=3 +5=8.

So that works as well.

Therefore, regardless of the values of a and b (as long as they satisfy the condition from f(-3)=2), f(3) is always 8.

Therefore, the answer is 8.

**Final Answer**
The value of $ f(3) $ is \boxed{8}.

To find the value of $ f(3) $, we're given the function:

$$
f(x) = ax^4 - bx^2 + x + 5
$$

and also the value of $ f(-3) = 2 $.

---

### Step 1: Use the given function to express $ f(-3) $

Plug $ x = -3 $ into the function:

$$
f(-3) = a(-3)^4 - b(-3)^2 + (-3) + 5
$$

Now compute each term:

- $ (-3)^4 = 81 $
- $ (-3)^2 = 9 $

So:

$$
f(-3) = 81a - 9b - 3 + 5 = 81a - 9b + 2
$$

We are told that $ f(-3) = 2 $, so:

$$
81a - 9b + 2 = 2
$$

Subtract 2 from both sides:

$$
81a - 9b = 0
$$

This gives a simple relationship between $ a $ and $ b $:

$$
81a - 9b = 0 \quad \Rightarrow \quad 9a - b = 0 \quad \Rightarrow \quad b = 9a
$$

---

### Step 2: Compute $ f(3) $

Now evaluate $ f(3) $ using the same function:

$$
f(3) = a(3)^4 - b(3)^2 + 3 + 5 = 81a - 9b + 3 + 5
$$

Simplify the constants:

$$
f(3) = 81a - 9b + 8
$$

From Step 1, we found that $ 81a - 9b = 0 $, so:

$$
f(3) = 0 + 8 = 8
$$

---

### Step 3: Verification

To ensure correctness, we observe the function has even and odd components:

- $ ax^4 - bx^2 $: even
- $ x $: odd
- $ 5 $: even

Let’s define:

$$
g(x) = ax^4 - bx^2 + 5 \quad \text{(even function)}
$$
$$
f(x) = g(x) + x
$$
$$
f(-x) = g(x) - x
$$

Then:

$$
f(x) + f(-x) = 2g(x)
$$

So for any $ x $, including $ x = 3 $:

$$
f(3) + f(-3) = 2g(3)
$$

We are told $ f(-3) = 2 $, and we are solving for $ f(3) $. So:

$$
f(3) + 2 = 2g(3)
$$

Now compute $ g(3) $:

$$
g(3) = a(3)^4 - b(3)^2 + 5 = 81a - 9b + 5 = 0 + 5 = 5
$$

So:

$$
f(3) + 2 = 2 \cdot 5 = 10 \quad \Rightarrow \quad f(3) = 8
$$

---

### Final Answer

$$
\boxed{8}
$$

Solution: Evaluating $f(x)$ for $x=3$ and $x=-3$, we have \[\left\{ \begin{aligned} f(3)& = a \cdot 3^4 - b \cdot 3^2 + 3 + 5, \\ f(-3) &= a \cdot (-3)^4 - b \cdot (-3)^2 + (-3) + 5. \end{aligned} \right.\]If we subtract the second equation from the first equation, all the terms but one cancel out, and we get \[f(3) - f(-3) = 3 - (-3) = 6.\]Thus, if $f(-3) = 2,$ then $f(3) = f(-3) + 6 = 2 + 6 = \boxed{8}.$


Explanation:
Okay, so I need to find the equation of a hyperbola given two foci, F1 at (10, 2) and F2 at (-16, 2), and the condition that the absolute difference of the distances from any point P on the hyperbola to the two foci is 24. The equation should be in the form (x - h)^2/a^2 - (y - k)^2/b^2 = 1, and then find h + k + a + b. Alright, let's start by recalling some properties of hyperbolas.

First, I remember that a hyperbola is defined as the set of all points where the absolute difference of the distances to two fixed points (the foci) is constant. That constant is equal to 2a, where a is the distance from the center to a vertex. So in this case, the given condition |PF1 - PF2| = 24 implies that 2a = 24, so a = 12. Got that part.

Next, I need to find the center of the hyperbola. The center is the midpoint between the two foci. Let me calculate that. The coordinates of the midpoint (h, k) can be found by averaging the x-coordinates and the y-coordinates of the foci.

For the x-coordinate: (10 + (-16))/2 = (-6)/2 = -3. For the y-coordinate: (2 + 2)/2 = 4/2 = 2. So the center is at (-3, 2). Therefore, h = -3 and k = 2. Let me note that down.

Now, to write the equation of the hyperbola, I need to know the distance between the center and each focus, which is c. The distance from the center to each focus is c. Then, for hyperbolas, the relationship between a, b, and c is c^2 = a^2 + b^2. So once I find c, I can solve for b.

First, let's find c. The distance between the two foci is 2c. Let me calculate the distance between F1 and F2. Since both points have the same y-coordinate (2), the distance between them is just the difference in x-coordinates. So distance = |10 - (-16)| = |26| = 26. Therefore, 2c = 26, so c = 13.

So now, we have a = 12, c = 13. Then we can find b using c^2 = a^2 + b^2. Let's plug in the numbers:

c^2 = 13^2 = 169

a^2 = 12^2 = 144

Therefore, 169 = 144 + b^2 => b^2 = 169 - 144 = 25 => b = 5. Since b is a length, we take the positive root.

Now, I need to figure out the orientation of the hyperbola. Since the foci are located along the horizontal line y = 2, which is horizontal. The midpoint is at (-3, 2). So the transverse axis is horizontal. Therefore, the standard form of the hyperbola is ((x - h)^2)/a^2 - ((y - k)^2)/b^2 = 1. Which is exactly the form given in the problem.

So plugging in the values we found:

h = -3, k = 2, a = 12, b = 5.

Therefore, h + k + a + b = (-3) + 2 + 12 + 5. Let me compute that step by step. -3 + 2 is -1. Then -1 + 12 is 11. Then 11 + 5 is 16. So the answer is 16? Wait, let me check again because sometimes I make arithmetic errors.

Wait: (-3) + 2 is -1. Then -1 + 12 is 11. Then 11 + 5 is indeed 16. Hmm. Let me check if I messed up any steps.

First, checking the midpoint: between (10, 2) and (-16, 2). The x-coordinate midpoint is (10 + (-16))/2 = (-6)/2 = -3. Y-coordinate is (2 + 2)/2 = 2. Correct. So center is (-3, 2). Correct.

Then, distance between foci: difference in x-coordinates is 10 - (-16) = 26, so distance is 26. Hence, 2c = 26 => c = 13. Correct. Then, since a = 12, and c^2 = a^2 + b^2 => 13^2 = 12^2 + b^2 => 169 = 144 + b^2 => b^2 = 25 => b = 5. Correct.

Equation is ((x - (-3))^2)/144 - ((y - 2)^2)/25 = 1, which simplifies to ((x + 3)^2)/144 - ((y - 2)^2)/25 = 1. So the form is correct. Then h = -3, k = 2, a = 12, b = 5. Then h + k + a + b = (-3) + 2 + 12 + 5. Let me check again:

-3 + 2 is -1. Then -1 + 12 is 11. Then 11 + 5 is 16. Yes, 16. But wait, the problem says "Find h + k + a + b." Let me check the problem again to make sure I didn't misread anything. The equation is supposed to be written as ((x - h)^2)/a^2 - ((y - k)^2)/b^2 = 1. So h is -3, k is 2, a is 12, and b is 5. So yes, adding them up: h + k + a + b = -3 + 2 + 12 + 5 = 16.

But let me check again because sometimes, depending on the hyperbola orientation, maybe I have to adjust something. Wait, the hyperbola is horizontal because the transverse axis is horizontal. So the standard form is indeed ((x - h)^2)/a^2 - ((y - k)^2)/b^2 = 1. So that's correct.

Alternatively, if the transverse axis was vertical, it would be ((y - k)^2)/a^2 - ((x - h)^2)/b^2 = 1, but since foci are on horizontal line, transverse axis is horizontal. So that's correct.

Therefore, I think the answer is 16. Wait, but let me check again the calculation of h + k + a + b.

h is -3, k is 2, so h + k is -1. Then a is 12, so -1 + 12 is 11. Then b is 5, so 11 + 5 is 16. Yeah, seems correct. I think that's right.

But just to make sure, let me think again. Suppose someone might think that since the hyperbola equation is ((x - h)^2)/a^2 - ((y - k)^2)/b^2 = 1, then h, k are coordinates of the center. So yes, center is (-3, 2), so h = -3, k = 2. Then a = 12, b = 5. So adding them up, -3 + 2 + 12 + 5.

Alternatively, maybe there's a mistake with the sign? For example, if the hyperbola was oriented differently. Wait, but the foci are on the horizontal line, so the hyperbola opens left and right. So the standard form is correct.

Alternatively, could the center be somewhere else? No, because the center is the midpoint of the foci. So I think that's correct.

Alternatively, maybe the problem is in the value of a? Let me check.

Given that |PF1 - PF2| = 2a. So 2a = 24, so a = 12. Correct. Since the definition of hyperbola is |PF1 - PF2| = 2a. So that's right.

So all steps seem correct. Therefore, the answer should be 16.

Wait, just to be thorough, maybe plug in the center coordinates into the equation? Wait, the center is (-3, 2). If we plug in x = -3, y = 2 into the equation, we get ((-3 + 3)^2)/144 - ((2 - 2)^2)/25 = 0 - 0 = 0, which is not equal to 1. But that's okay because the center is not on the hyperbola. The hyperbola doesn't include the center. So that's fine.

Alternatively, maybe check a vertex? The vertices are located at a distance a from the center along the transverse axis. Since the transverse axis is horizontal, the vertices are at (h ± a, k). So that would be (-3 + 12, 2) = (9, 2) and (-3 - 12, 2) = (-15, 2). Let me check if these points satisfy the original condition |PF1 - PF2| = 24.

Take the vertex at (9, 2). Let's compute PF1 and PF2.

PF1 is the distance from (9, 2) to F1 (10, 2): sqrt[(9 - 10)^2 + (2 - 2)^2] = sqrt[1 + 0] = 1.

PF2 is the distance from (9, 2) to F2 (-16, 2): sqrt[(9 - (-16))^2 + (2 - 2)^2] = sqrt[(25)^2 + 0] = 25.

So |1 - 25| = 24. Which is correct. Similarly, check the other vertex (-15, 2). Distance to F1 (10, 2): sqrt[(-15 - 10)^2 + (2 - 2)^2] = sqrt[(-25)^2] = 25. Distance to F2 (-16, 2): sqrt[(-15 - (-16))^2 + (2 - 2)^2] = sqrt[(1)^2] = 1. So |25 - 1| = 24. Correct. So that checks out. Therefore, the values for a, h, k, and b seem correct.

Another check: the distance from the center to each focus is c = 13. The center is at (-3, 2). Let's check distance from center to F1 (10, 2). That is sqrt[(10 - (-3))^2 + (2 - 2)^2] = sqrt[(13)^2] = 13. Correct. Similarly for F2 (-16, 2): sqrt[(-16 - (-3))^2 + (2 - 2)^2] = sqrt[(-13)^2] = 13. Correct. So c is correct. Then, since a = 12, and c^2 = a^2 + b^2, we have b^2 = c^2 - a^2 = 169 - 144 = 25, so b = 5. Correct.

So all the numbers check out. Therefore, h + k + a + b = -3 + 2 + 12 + 5 = 16. Therefore, the answer is 16.

I think that's solid. I don't see any mistakes in my reasoning. So I feel confident that the answer is 16.

**Final Answer**
\boxed{16}

To solve the problem, we are given two foci of a hyperbola:
- $ F_1 = (10, 2) $
- $ F_2 = (-16, 2) $

The set of all points $ P $ satisfying the condition
$$
|PF_1 - PF_2| = 24
$$
forms a hyperbola. We are asked to find the sum $ h + k + a + b $, where the hyperbola is written in the standard form:
$$
\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1
$$

---

### Step 1: Find the Center of the Hyperbola

The **center** of a hyperbola is the **midpoint** of its foci. The midpoint $ (h, k) $ is given by:
$$
h = \frac{10 + (-16)}{2} = \frac{-6}{2} = -3, \quad
k = \frac{2 + 2}{2} = \frac{4}{2} = 2
$$
So the center is at $ (-3, 2) $.

---

### Step 2: Determine $ a $ from the Given Difference

The condition $ |PF_1 - PF_2| = 24 $ defines the constant difference in distances from a point on the hyperbola to the foci, which is equal to $ 2a $. So:
$$
2a = 24 \Rightarrow a = 12
$$

---

### Step 3: Calculate the Distance from the Center to a Focus (i.e., $ c $)

The distance between the foci is:
$$
\text{Distance} = |10 - (-16)| = |26| = 26
$$
So the distance from the center to each focus is:
$$
c = \frac{26}{2} = 13
$$

---

### Step 4: Find $ b $ Using the Hyperbola Identity

For a hyperbola, the relationship between $ a $, $ b $, and $ c $ is:
$$
c^2 = a^2 + b^2
$$
Substitute the known values:
$$
13^2 = 12^2 + b^2 \Rightarrow 169 = 144 + b^2 \Rightarrow b^2 = 25 \Rightarrow b = 5
$$

---

### Step 5: Verify the Orientation and Write the Equation

Since the foci have the same $ y $-coordinate, the transverse axis is **horizontal**, so the standard form of the hyperbola is:
$$
\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1
$$
Substituting $ h = -3 $, $ k = 2 $, $ a = 12 $, and $ b = 5 $, we get:
$$
\frac{(x + 3)^2}{144} - \frac{(y - 2)^2}{25} = 1
$$

---

### Step 6: Compute the Final Answer

We are asked to compute:
$$
h + k + a + b = (-3) + 2 + 12 + 5 = 16
$$

---

### Final Answer

$$
\boxed{16}
$$

[Assistant]
Explanation:
Okay, so I need to find the equation of a hyperbola given two foci, F1 at (10, 2) and F2 at (-16, 2), and the condition that the absolute difference of the distances from any point P on the hyperbola to the two foci is 24. The equation should be in the form (x - h)^2/a^2 - (y - k)^2/b^2 = 1, and then find h + k + a + b. Alright, let's start by recalling some properties of hyperbolas.

First, I remember that a hyperbola is defined as the set of all points where the absolute difference of the distances to two fixed points (the foci) is constant. That constant is equal to 2a, where a is the distance from the center to a vertex. So in this case, the given condition |PF1 - PF2| = 24 implies that 2a = 24, so a = 12. Got that part.

Next, I need to find the center of the hyperbola. The center is the midpoint between the two foci. Let me calculate that. The coordinates of the midpoint (h, k) can be found by averaging the x-coordinates and the y-coordinates of the foci.

For the x-coordinate: (10 + (-16))/2 = (-6)/2 = -3. For the y-coordinate: (2 + 2)/2 = 4/2 = 2. So the center is at (-3, 2). Therefore, h = -3 and k = 2. Let me note that down.

Now, to write the equation of the hyperbola, I need to know the distance between the center and each focus, which is c. The distance from the center to each focus is c. Then, for hyperbolas, the relationship between a, b, and c is c^2 = a^2 + b^2. So once I find c, I can solve for b.

First, let's find c. The distance between the two foci is 2c. Let me calculate the distance between F1 and F2. Since both points have the same y-coordinate (2), the distance between them is just the difference in x-coordinates. So distance = |10 - (-16)| = |26| = 26. Therefore, 2c = 26, so c = 13.

So now, we have a = 12, c = 13. Then we can find b using c^2 = a^2 + b^2. Let's plug in the numbers:

c^2 = 13^2 = 169

a^2 = 12^2 = 144

Therefore, 169 = 144 + b^2 => b^2 = 169 - 144 = 25 => b = 5. Since b is a length, we take the positive root.

Now, I need to figure out the orientation of the hyperbola. Since the foci are located along the horizontal line y = 2, which is horizontal. The midpoint is at (-3, 2). So the transverse axis is horizontal. Therefore, the standard form of the hyperbola is ((x - h)^2)/a^2 - ((y - k)^2)/b^2 = 1. Which is exactly the form given in the problem.

So plugging in the values we found:

h = -3, k = 2, a = 12, b = 5.

Therefore, h + k + a + b = (-3) + 2 + 12 + 5. Let me compute that step by step. -3 + 2 is -1. Then -1 + 12 is 11. Then 11 + 5 is 16. So the answer is 16? Wait, let me check again because sometimes I make arithmetic errors.

Wait: (-3) + 2 is -1. Then -1 + 12 is 11. Then 11 + 5 is indeed 16. Hmm. Let me check if I messed up any steps.

First, checking the midpoint: between (10, 2) and (-16, 2). The x-coordinate midpoint is (10 + (-16))/2 = (-6)/2 = -3. Y-coordinate is (2 + 2)/2 = 2. Correct. So center is (-3, 2). Correct.

Then, distance between foci: difference in x-coordinates is 10 - (-16) = 26, so distance is 26. Hence, 2c = 26 => c = 13. Correct. Then, since a = 12, and c^2 = a^2 + b^2 => 13^2 = 12^2 + b^2 => 169 = 144 + b^2 => b^2 = 25 => b = 5. Correct.

Equation is ((x - (-3))^2)/144 - ((y - 2)^2)/25 = 1, which simplifies to ((x + 3)^2)/144 - ((y - 2)^2)/25 = 1. So the form is correct. Then h = -3, k = 2, a = 12, b = 5. Then h + k + a + b = (-3) + 2 + 12 + 5. Let me check again:

-3 + 2 is -1. Then -1 + 12 is 11. Then 11 + 5 is 16. Yes, 16. But wait, the problem says "Find h + k + a + b." Let me check the problem again to make sure I didn't misread anything. The equation is supposed to be written as ((x - h)^2)/a^2 - ((y - k)^2)/b^2 = 1. So h is -3, k is 2, a is 12, and b is 5. So yes, adding them up: h + k + a + b = -3 + 2 + 12 + 5 = 16.

But let me check again because sometimes, depending on the hyperbola orientation, maybe I have to adjust something. Wait, the hyperbola is horizontal because the transverse axis is horizontal. So the standard form is indeed ((x - h)^2)/a^2 - ((y - k)^2)/b^2 = 1. So that's correct.

Alternatively, if the transverse axis was vertical, it would be ((y - k)^2)/a^2 - ((x - h)^2)/b^2 = 1, but since foci are on horizontal line, transverse axis is horizontal. So that's correct.

Therefore, I think the answer is 16. Wait, but let me check again the calculation of h + k + a + b.

h is -3, k is 2, so h + k is -1. Then a is 12, so -1 + 12 is 11. Then b is 5, so 11 + 5 is 16. Yeah, seems correct. I think that's right.

But just to make sure, let me think again. Suppose someone might think that since the hyperbola equation is ((x - h)^2)/a^2 - ((y - k)^2)/b^2 = 1, then h, k are coordinates of the center. So yes, center is (-3, 2), so h = -3, k = 2. Then a = 12, b = 5. So adding them up, -3 + 2 + 12 + 5.

Alternatively, maybe there's a mistake with the sign? For example, if the hyperbola was oriented differently. Wait, but the foci are on the horizontal line, so the hyperbola opens left and right. So the standard form is correct.

Alternatively, could the center be somewhere else? No, because the center is the midpoint of the foci. So I think that's correct.

Alternatively, maybe the problem is in the value of a? Let me check.

Given that |PF1 - PF2| = 2a. So 2a = 24, so a = 12. Correct. Since the definition of hyperbola is |PF1 - PF2| = 2a. So that's right.

So all steps seem correct. Therefore, the answer should be 16.

Wait, just to be thorough, maybe plug in the center coordinates into the equation? Wait, the center is (-3, 2). If we plug in x = -3, y = 2 into the equation, we get ((-3 + 3)^2)/144 - ((2 - 2)^2)/25 = 0 - 0 = 0, which is not equal to 1. But that's okay because the center is not on the hyperbola. The hyperbola doesn't include the center. So that's fine.

Alternatively, maybe check a vertex? The vertices are located at a distance a from the center along the transverse axis. Since the transverse axis is horizontal, the vertices are at (h ± a, k). So that would be (-3 + 12, 2) = (9, 2) and (-3 - 12, 2) = (-15, 2). Let me check if these points satisfy the original condition |PF1 - PF2| = 24.

Take the vertex at (9, 2). Let's compute PF1 and PF2.

PF1 is the distance from (9, 2) to F1 (10, 2): sqrt[(9 - 10)^2 + (2 - 2)^2] = sqrt[1 + 0] = 1.

PF2 is the distance from (9, 2) to F2 (-16, 2): sqrt[(9 - (-16))^2 + (2 - 2)^2] = sqrt[(25)^2 + 0] = 25.

So |1 - 25| = 24. Which is correct. Similarly, check the other vertex (-15, 2). Distance to F1 (10, 2): sqrt[(-15 - 10)^2 + (2 - 2)^2] = sqrt[(-25)^2] = 25. Distance to F2 (-16, 2): sqrt[(-15 - (-16))^2 + (2 - 2)^2] = sqrt[(1)^2] = 1. So |25 - 1| = 24. Correct. So that checks out. Therefore, the values for a, h, k, and b seem correct.

Another check: the distance from the center to each focus is c = 13. The center is at (-3, 2). Let's check distance from center to F1 (10, 2). That is sqrt[(10 - (-3))^2 + (2 - 2)^2] = sqrt[(13)^2] = 13. Correct. Similarly for F2 (-16, 2): sqrt[(-16 - (-3))^2 + (2 - 2)^2] = sqrt[(-13)^2] = 13. Correct. So c is correct. Then, since a = 12, and c^2 = a^2 + b^2, we have b^2 = c^2 - a^2 = 169 - 144 = 25, so b = 5. Correct.

So all the numbers check out. Therefore, h + k + a + b = -3 + 2 + 12 + 5 = 16. Therefore, the answer is 16.

I think that's solid. I don't see any mistakes in my reasoning. So I feel confident that the answer is 16.

**Final Answer**
\boxed{16}

To solve the problem, we are given two foci of a hyperbola:
- $ F_1 = (10, 2) $
- $ F_2 = (-16, 2) $

The set of all points $ P $ satisfying the condition
$$
|PF_1 - PF_2| = 24
$$
forms a hyperbola. We are asked to find the sum $ h + k + a + b $, where the hyperbola is written in the standard form:
$$
\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1
$$

---

### Step 1: Find the Center of the Hyperbola

The **center** of a hyperbola is the **midpoint** of its foci. The midpoint $ (h, k) $ is given by:
$$
h = \frac{10 + (-16)}{2} = \frac{-6}{2} = -3, \quad
k = \frac{2 + 2}{2} = \frac{4}{2} = 2
$$
So the center is at $ (-3, 2) $.

---

### Step 2: Determine $ a $ from the Given Difference

The condition $ |PF_1 - PF_2| = 24 $ defines the constant difference in distances from a point on the hyperbola to the foci, which is equal to $ 2a $. So:
$$
2a = 24 \Rightarrow a = 12
$$

---

### Step 3: Calculate the Distance from the Center to a Focus (i.e., $ c $)

The distance between the foci is:
$$
\text{Distance} = |10 - (-16)| = |26| = 26
$$
So the distance from the center to each focus is:
$$
c = \frac{26}{2} = 13
$$

---

### Step 4: Find $ b $ Using the Hyperbola Identity

For a hyperbola, the relationship between $ a $, $ b $, and $ c $ is:
$$
c^2 = a^2 + b^2
$$
Substitute the known values:
$$
13^2 = 12^2 + b^2 \Rightarrow 169 = 144 + b^2 \Rightarrow b^2 = 25 \Rightarrow b = 5
$$

---

### Step 5: Verify the Orientation and Write the Equation

Since the foci have the same $ y $-coordinate, the transverse axis is **horizontal**, so the standard form of the hyperbola is:
$$
\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1
$$
Substituting $ h = -3 $, $ k = 2 $, $ a = 12 $, and $ b = 5 $, we get:
$$
\frac{(x + 3)^2}{144} - \frac{(y - 2)^2}{25} = 1
$$

---

### Step 6: Compute the Final Answer

We are asked to compute:
$$
h + k + a + b = (-3) + 2 + 12 + 5 = 16
$$

---

### Final Answer

$$
\boxed{16}
$$

Solution:
The center of the hyperbola is the midpoint of $\overline{F_1 F_2},$ which is $(-3,2).$  Thus, $h = -3$ and $k = 2.$

Also, $2a = 24,$ so $a = 12.$  The distance between the foci is $2c = 26,$ so $c = 13.$  Then $b^2 = c^2 - a^2 = 169 - 144 = 25,$ so $b = 5.$

Hence, $h + k + a + b = (-3) + 2 + 12 + 5 = \boxed{16}.$


Explanation:
Okay, so I need to divide 413 base 5 by 2 base 5 and give the answer in base 5. Hmm, let me think about how to approach this. I remember that dividing numbers in different bases can be tricky, but maybe I can convert them to base 10 first, do the division there, and then convert the result back to base 5? Let me try that.

First, let me convert 413 base 5 to decimal. Each digit represents a power of 5. Starting from the right, the digits are 3, 1, 4. So the rightmost digit is 5^0, then 5^1, then 5^2. Let me calculate:

The rightmost digit is 3 * 5^0 = 3 * 1 = 3.

The middle digit is 1 * 5^1 = 1 * 5 = 5.

The leftmost digit is 4 * 5^2 = 4 * 25 = 100.

Adding those up: 100 + 5 + 3 = 108. So 413 base 5 is 108 in decimal.

Now, the divisor is 2 base 5. Converting that to decimal, it's just 2 * 5^0 = 2. So 2 base 5 is 2 in decimal.

So the division in decimal is 108 divided by 2, which is 54. Now, I need to convert 54 back to base 5.

To convert 54 from decimal to base 5, I need to find the coefficients for powers of 5. Let me recall that 5^0 = 1, 5^1 = 5, 5^2 = 25, 5^3 = 125. Since 125 is larger than 54, the highest power needed is 5^2 = 25.

So, let's divide 54 by 25. 25 * 2 = 50, which is less than 54. 25 * 3 = 75, which is more. So the coefficient for 5^2 is 2. Then subtract 50 from 54, which leaves 4.

Next, move to 5^1 (which is 5). 4 divided by 5 is 0 with remainder 4. So the coefficient for 5^1 is 0.

Then 5^0 is 1. The remainder is 4, so coefficient for 5^0 is 4.

Putting it all together: 2 * 5^2 + 0 * 5^1 + 4 * 5^0, so the base 5 number is 204. Let me check that. 2*25 + 0*5 + 4*1 = 50 + 0 + 4 = 54. Yep, that's correct.

Therefore, 413 base 5 divided by 2 base 5 is 204 base 5. Wait, but maybe I should verify this by performing the division directly in base 5 to make sure I didn't make a mistake? Because sometimes converting back and forth can have errors.

Let me try doing the division in base 5. Let me recall how long division works in base 5.

First, let me write down the numbers. The dividend is 413 base 5, divisor is 2 base 5. Let me set it up like regular long division.

So, 2 base 5 ) 413 base 5. Let me think about how this works. Since the divisor is 2, and we're working in base 5, I need to figure out how many times 2 goes into each digit of the dividend.

Starting from the leftmost digit, which is 4. So, how many times does 2 base 5 go into 4 base 5? In decimal, 4 base 5 is 4, and 2 base 5 is 2. So, 4 divided by 2 is 2. So we write 2 as the first digit of the quotient.

Then, we multiply 2 (base 5) by 2 (base 5), which is 4 (base 5). Subtract that from 4 (base 5), which gives 0. Bring down the next digit, which is 1. Now, we have 01 (base 5). How many times does 2 base 5 go into 1 base 5? Zero times. So we write 0 as the next digit in the quotient. Multiply 0 by 2, get 0, subtract from 1, and bring down the next digit, which is 3. Now we have 13 base 5.

So, how many times does 2 base 5 go into 13 base 5? Let me convert 13 base 5 to decimal. 1*5 + 3 = 8. Divisor is 2 base 5 which is 2 decimal. So 8 divided by 2 is 4. So 4 times. Let me check if 4 * 2 base 5 is equal to 13 base 5? Let me compute 4 * 2 in base 5. 4 * 2 = 8 decimal. 8 in base 5 is 13 (since 1*5 + 3 = 8). So yes, so that works. Therefore, we write 4 as the next digit in the quotient. Then subtract 13 base 5 from 13 base 5, which gives 0. So there's no remainder.

Therefore, the quotient is 2 0 4, which is 204 base 5. So that matches my previous result. Therefore, the answer is 204 base 5. That seems to check out.

But just to be thorough, let me verify this another way. Let me multiply 204 base 5 by 2 base 5 and see if I get 413 base 5.

First, converting 204 base 5 to decimal: 2*25 + 0*5 + 4*1 = 50 + 0 + 4 = 54. Then multiplying by 2 gives 108. Which is 413 base 5. So yes, that's correct.

Alternatively, doing the multiplication directly in base 5. Let's try multiplying 204 base 5 by 2 base 5.

Starting from the rightmost digit: 4 * 2. 4 base 5 times 2 base 5. In decimal, that's 4*2=8. Convert back to base 5: 8 divided by 5 is 1 with a remainder of 3. So write down 3 and carry over 1.

Next digit: 0 * 2 + carryover 1. 0*2 is 0 + 1 = 1. So write down 1.

Next digit: 2 * 2 = 4. Write down 4. So the product is 4 1 3, which is 413 base 5. Perfect, that's exactly the original number. So that confirms that 204 base 5 multiplied by 2 base 5 is indeed 413 base 5, so the division is correct.

Therefore, I can be confident that the answer is 204 base 5.

But just to recap the steps, in case I missed anything:

First, converting both numbers to decimal: 413_5 = 108, 2_5 = 2. 108 divided by 2 is 54. Convert 54 to base 5: 2*25 + 0*5 + 4*1 = 204_5. Then verified by performing division in base 5 and multiplication back to confirm. All steps check out.

Alternatively, if I didn't want to convert to decimal, could I do the division directly in base 5? Let me try that again as an exercise.

So, dividing 413_5 by 2_5. Let me recall that division in base 5 is similar to base 10, but all the operations are done in base 5.

Let me write it out:

           ______
2_5 ) 413_5

First, divide the leftmost digit, which is 4_5. Since 2_5 goes into 4_5 twice (since 2*2=4), we write 2 in the quotient.

Multiply 2_5 by 2_5: 2*2=4_5. Subtract 4_5 from 4_5: 0_5. Bring down the next digit, which is 1_5. Now, we have 01_5.

Now, 2_5 goes into 1_5 how many times? 0 times. So we write 0 in the quotient. Multiply 0 by 2_5, get 0_5. Subtract 0_5 from 1_5, get 1_5. Bring down the next digit, 3_5, making it 13_5.

Now, divide 13_5 by 2_5. Let me figure out how many times 2_5 goes into 13_5. Let's think in base 5. Let me recall that 2_5 times 2_5 is 4_5, 2_5 times 3_5 is 11_5 (since 2*3=6 decimal, which is 1*5 + 1). But 13_5 is 8 decimal. So 2_5 times 4_5 is 13_5? Wait, 4_5 times 2_5.

Wait, let me do that multiplication. 4_5 * 2_5. In decimal, 4*2=8, which is 13_5. So yes, 4_5 * 2_5 = 13_5. Therefore, 2_5 goes into 13_5 exactly four times. So write 4 in the quotient. Then subtract 13_5 from 13_5, which gives 0. Therefore, the quotient is 2 0 4_5, which is 204_5. Same result.

Therefore, regardless of the method, I get 204_5. So I think that's solid.

Wait, just another thought: when I did the initial division in base 5, when I had 1_5 after bringing down the 3, I had to think about how many times 2_5 goes into 13_5. But maybe another way to think about it is in base 5, so maybe I can do the division without converting to decimal. Let me try that.

So, 2_5 into 13_5. Let me think of it as a base 5 number. Let me note that 2_5 * 1_5 = 2_5, 2_5 * 2_5 = 4_5, 2_5 * 3_5 = 11_5, 2_5 * 4_5 = 13_5. Therefore, 2_5 * 4_5 = 13_5. Therefore, the answer is 4. So that's how I knew to put 4 there. So that's correct.

Therefore, all steps are correct. Therefore, the answer is 204 base 5.

Alternatively, maybe someone could make a mistake in conversion? For example, converting 413 base 5 to decimal. Let me check that again. 4*25 + 1*5 + 3*1 = 100 + 5 + 3 = 108. Correct. Then 108 divided by 2 is 54. Then converting 54 to base 5.

Another way to convert 54 to base 5: divide by 5, record remainders.

54 divided by 5 is 10 with remainder 4.

10 divided by 5 is 2 with remainder 0.

2 divided by 5 is 0 with remainder 2.

So reading the remainders from last to first: 2, 0, 4, which is 204 base 5. Perfect, that's consistent.

Therefore, multiple methods confirm that the result is 204 base 5.

I think that's thorough. I can't think of any other way this could be approached. Maybe someone might try to do it all in base 5 without converting, but the methods I did are standard. So I think this should be correct.

**Final Answer**
The quotient of $413_5 \div 2_5$ in base 5 is \boxed{204_5}.

To solve the problem of dividing $413_5 \div 2_5$ and expressing the result in base 5, we follow a systematic approach, converting the numbers into decimal, performing the division, and then converting the result back to base 5.

---

### Step 1: Convert $413_5$ to Decimal

We interpret each digit of $413_5$ as follows:

- Rightmost digit: $3 \times 5^0 = 3$
- Middle digit: $1 \times 5^1 = 5$
- Leftmost digit: $4 \times 5^2 = 100$

Adding these values:

$$
3 + 5 + 100 = 108
$$

So, $413_5 = 108_{10}$

---

### Step 2: Convert $2_5$ to Decimal

$$
2_5 = 2_{10}
$$

---

### Step 3: Perform the Division in Decimal

$$
108 \div 2 = 54
$$

---

### Step 4: Convert $54_{10}$ to Base 5

We divide by the largest powers of 5:

- $5^3 = 125$ is too large, so the highest relevant power is $5^2 = 25$
- $54 \div 25 = 2$ with remainder $4$
- $4 \div 5 = 0$ with remainder $4$
- $4 \div 1 = 4$

So the coefficients are:

- $2 \times 25$
- $0 \times 5$
- $4 \times 1$

Thus, the base 5 number is:
$$
204_5
$$

---

### Step 5: Verification by Direct Base 5 Division

To ensure accuracy, we perform the division directly in base 5:

- $413_5 \div 2_5$
- Start by dividing leftmost digit: $4_5 \div 2_5 = 2_5$, write 2 in the quotient
- Multiply $2_5 \times 2_5 = 4_5$, subtract from $4_5$ ⇒ remainder 0
- Bring down 1 ⇒ now divide $1_5 \div 2_5 = 0$, write 0 in the quotient
- Bring down 3 ⇒ now divide $13_5 \div 2_5 = 4_5$, since $2_5 \times 4_5 = 13_5$

So the quotient is $204_5$, and there is no remainder.

---

### Final Answer

$$
\boxed{204_5}
$$

[Assistant]
Explanation:
Okay, so I need to divide 413 base 5 by 2 base 5 and give the answer in base 5. Hmm, let me think about how to approach this. I remember that dividing numbers in different bases can be tricky, but maybe I can convert them to base 10 first, do the division there, and then convert the result back to base 5? Let me try that.

First, let me convert 413 base 5 to decimal. Each digit represents a power of 5. Starting from the right, the digits are 3, 1, 4. So the rightmost digit is 5^0, then 5^1, then 5^2. Let me calculate:

The rightmost digit is 3 * 5^0 = 3 * 1 = 3.

The middle digit is 1 * 5^1 = 1 * 5 = 5.

The leftmost digit is 4 * 5^2 = 4 * 25 = 100.

Adding those up: 100 + 5 + 3 = 108. So 413 base 5 is 108 in decimal.

Now, the divisor is 2 base 5. Converting that to decimal, it's just 2 * 5^0 = 2. So 2 base 5 is 2 in decimal.

So the division in decimal is 108 divided by 2, which is 54. Now, I need to convert 54 back to base 5.

To convert 54 from decimal to base 5, I need to find the coefficients for powers of 5. Let me recall that 5^0 = 1, 5^1 = 5, 5^2 = 25, 5^3 = 125. Since 125 is larger than 54, the highest power needed is 5^2 = 25.

So, let's divide 54 by 25. 25 * 2 = 50, which is less than 54. 25 * 3 = 75, which is more. So the coefficient for 5^2 is 2. Then subtract 50 from 54, which leaves 4.

Next, move to 5^1 (which is 5). 4 divided by 5 is 0 with remainder 4. So the coefficient for 5^1 is 0.

Then 5^0 is 1. The remainder is 4, so coefficient for 5^0 is 4.

Putting it all together: 2 * 5^2 + 0 * 5^1 + 4 * 5^0, so the base 5 number is 204. Let me check that. 2*25 + 0*5 + 4*1 = 50 + 0 + 4 = 54. Yep, that's correct.

Therefore, 413 base 5 divided by 2 base 5 is 204 base 5. Wait, but maybe I should verify this by performing the division directly in base 5 to make sure I didn't make a mistake? Because sometimes converting back and forth can have errors.

Let me try doing the division in base 5. Let me recall how long division works in base 5.

First, let me write down the numbers. The dividend is 413 base 5, divisor is 2 base 5. Let me set it up like regular long division.

So, 2 base 5 ) 413 base 5. Let me think about how this works. Since the divisor is 2, and we're working in base 5, I need to figure out how many times 2 goes into each digit of the dividend.

Starting from the leftmost digit, which is 4. So, how many times does 2 base 5 go into 4 base 5? In decimal, 4 base 5 is 4, and 2 base 5 is 2. So, 4 divided by 2 is 2. So we write 2 as the first digit of the quotient.

Then, we multiply 2 (base 5) by 2 (base 5), which is 4 (base 5). Subtract that from 4 (base 5), which gives 0. Bring down the next digit, which is 1. Now, we have 01 (base 5). How many times does 2 base 5 go into 1 base 5? Zero times. So we write 0 as the next digit in the quotient. Multiply 0 by 2, get 0, subtract from 1, and bring down the next digit, which is 3. Now we have 13 base 5.

So, how many times does 2 base 5 go into 13 base 5? Let me convert 13 base 5 to decimal. 1*5 + 3 = 8. Divisor is 2 base 5 which is 2 decimal. So 8 divided by 2 is 4. So 4 times. Let me check if 4 * 2 base 5 is equal to 13 base 5? Let me compute 4 * 2 in base 5. 4 * 2 = 8 decimal. 8 in base 5 is 13 (since 1*5 + 3 = 8). So yes, so that works. Therefore, we write 4 as the next digit in the quotient. Then subtract 13 base 5 from 13 base 5, which gives 0. So there's no remainder.

Therefore, the quotient is 2 0 4, which is 204 base 5. So that matches my previous result. Therefore, the answer is 204 base 5. That seems to check out.

But just to be thorough, let me verify this another way. Let me multiply 204 base 5 by 2 base 5 and see if I get 413 base 5.

First, converting 204 base 5 to decimal: 2*25 + 0*5 + 4*1 = 50 + 0 + 4 = 54. Then multiplying by 2 gives 108. Which is 413 base 5. So yes, that's correct.

Alternatively, doing the multiplication directly in base 5. Let's try multiplying 204 base 5 by 2 base 5.

Starting from the rightmost digit: 4 * 2. 4 base 5 times 2 base 5. In decimal, that's 4*2=8. Convert back to base 5: 8 divided by 5 is 1 with a remainder of 3. So write down 3 and carry over 1.

Next digit: 0 * 2 + carryover 1. 0*2 is 0 + 1 = 1. So write down 1.

Next digit: 2 * 2 = 4. Write down 4. So the product is 4 1 3, which is 413 base 5. Perfect, that's exactly the original number. So that confirms that 204 base 5 multiplied by 2 base 5 is indeed 413 base 5, so the division is correct.

Therefore, I can be confident that the answer is 204 base 5.

But just to recap the steps, in case I missed anything:

First, converting both numbers to decimal: 413_5 = 108, 2_5 = 2. 108 divided by 2 is 54. Convert 54 to base 5: 2*25 + 0*5 + 4*1 = 204_5. Then verified by performing division in base 5 and multiplication back to confirm. All steps check out.

Alternatively, if I didn't want to convert to decimal, could I do the division directly in base 5? Let me try that again as an exercise.

So, dividing 413_5 by 2_5. Let me recall that division in base 5 is similar to base 10, but all the operations are done in base 5.

Let me write it out:

           ______
2_5 ) 413_5

First, divide the leftmost digit, which is 4_5. Since 2_5 goes into 4_5 twice (since 2*2=4), we write 2 in the quotient.

Multiply 2_5 by 2_5: 2*2=4_5. Subtract 4_5 from 4_5: 0_5. Bring down the next digit, which is 1_5. Now, we have 01_5.

Now, 2_5 goes into 1_5 how many times? 0 times. So we write 0 in the quotient. Multiply 0 by 2_5, get 0_5. Subtract 0_5 from 1_5, get 1_5. Bring down the next digit, 3_5, making it 13_5.

Now, divide 13_5 by 2_5. Let me figure out how many times 2_5 goes into 13_5. Let's think in base 5. Let me recall that 2_5 times 2_5 is 4_5, 2_5 times 3_5 is 11_5 (since 2*3=6 decimal, which is 1*5 + 1). But 13_5 is 8 decimal. So 2_5 times 4_5 is 13_5? Wait, 4_5 times 2_5.

Wait, let me do that multiplication. 4_5 * 2_5. In decimal, 4*2=8, which is 13_5. So yes, 4_5 * 2_5 = 13_5. Therefore, 2_5 goes into 13_5 exactly four times. So write 4 in the quotient. Then subtract 13_5 from 13_5, which gives 0. Therefore, the quotient is 2 0 4_5, which is 204_5. Same result.

Therefore, regardless of the method, I get 204_5. So I think that's solid.

Wait, just another thought: when I did the initial division in base 5, when I had 1_5 after bringing down the 3, I had to think about how many times 2_5 goes into 13_5. But maybe another way to think about it is in base 5, so maybe I can do the division without converting to decimal. Let me try that.

So, 2_5 into 13_5. Let me think of it as a base 5 number. Let me note that 2_5 * 1_5 = 2_5, 2_5 * 2_5 = 4_5, 2_5 * 3_5 = 11_5, 2_5 * 4_5 = 13_5. Therefore, 2_5 * 4_5 = 13_5. Therefore, the answer is 4. So that's how I knew to put 4 there. So that's correct.

Therefore, all steps are correct. Therefore, the answer is 204 base 5.

Alternatively, maybe someone could make a mistake in conversion? For example, converting 413 base 5 to decimal. Let me check that again. 4*25 + 1*5 + 3*1 = 100 + 5 + 3 = 108. Correct. Then 108 divided by 2 is 54. Then converting 54 to base 5.

Another way to convert 54 to base 5: divide by 5, record remainders.

54 divided by 5 is 10 with remainder 4.

10 divided by 5 is 2 with remainder 0.

2 divided by 5 is 0 with remainder 2.

So reading the remainders from last to first: 2, 0, 4, which is 204 base 5. Perfect, that's consistent.

Therefore, multiple methods confirm that the result is 204 base 5.

I think that's thorough. I can't think of any other way this could be approached. Maybe someone might try to do it all in base 5 without converting, but the methods I did are standard. So I think this should be correct.

**Final Answer**
The quotient of $413_5 \div 2_5$ in base 5 is \boxed{204_5}.

To solve the problem of dividing $413_5 \div 2_5$ and expressing the result in base 5, we follow a systematic approach, converting the numbers into decimal, performing the division, and then converting the result back to base 5.

---

### Step 1: Convert $413_5$ to Decimal

We interpret each digit of $413_5$ as follows:

- Rightmost digit: $3 \times 5^0 = 3$
- Middle digit: $1 \times 5^1 = 5$
- Leftmost digit: $4 \times 5^2 = 100$

Adding these values:

$$
3 + 5 + 100 = 108
$$

So, $413_5 = 108_{10}$

---

### Step 2: Convert $2_5$ to Decimal

$$
2_5 = 2_{10}
$$

---

### Step 3: Perform the Division in Decimal

$$
108 \div 2 = 54
$$

---

### Step 4: Convert $54_{10}$ to Base 5

We divide by the largest powers of 5:

- $5^3 = 125$ is too large, so the highest relevant power is $5^2 = 25$
- $54 \div 25 = 2$ with remainder $4$
- $4 \div 5 = 0$ with remainder $4$
- $4 \div 1 = 4$

So the coefficients are:

- $2 \times 25$
- $0 \times 5$
- $4 \times 1$

Thus, the base 5 number is:
$$
204_5
$$

---

### Step 5: Verification by Direct Base 5 Division

To ensure accuracy, we perform the division directly in base 5:

- $413_5 \div 2_5$
- Start by dividing leftmost digit: $4_5 \div 2_5 = 2_5$, write 2 in the quotient
- Multiply $2_5 \times 2_5 = 4_5$, subtract from $4_5$ ⇒ remainder 0
- Bring down 1 ⇒ now divide $1_5 \div 2_5 = 0$, write 0 in the quotient
- Bring down 3 ⇒ now divide $13_5 \div 2_5 = 4_5$, since $2_5 \times 4_5 = 13_5$

So the quotient is $204_5$, and there is no remainder.

---

### Final Answer

$$
\boxed{204_5}
$$

Solution:
We may carry out long division in base 5 just as in base 10. We have  \[
\begin{array}{c|ccc}
\multicolumn{2}{r}{2} & 0 & 4 \\
\cline{2-4}
2 & 4 & 1 & 3 \\
\multicolumn{2}{r}{4} & \downarrow & \\ \cline{2-2}
\multicolumn{2}{r}{0} & 1 & \\
\multicolumn{2}{r}{} & 0 & \downarrow \\ \cline{3-3}
\multicolumn{2}{r}{} & 1 & 3 \\
\multicolumn{2}{r}{} & 1 & 3 \\ \cline{3-4}
\multicolumn{2}{r}{} & & 0
\end{array}
\]for a quotient of $\boxed{204_5}$. Note that in the above calculation we have used that $13_5$ divided by $2_5$ is $4_5$, which follows from $4_5\times2_5=8_{10}=13_5$.