Explanation:
Okay, so I need to find a two-digit integer AB such that when it's cubed, the result is 912,673. Then, I have to find the sum of its digits A and B. Alright, let me think about how to approach this.

First, let me recall that AB is a two-digit number. That means it's actually 10*A + B, right? So, for example, if A was 1 and B was 2, the number would be 12. But here, we don't know A and B yet. The problem is telling me that when I cube this number, I get 912,673. So, mathematically, that would be:

(10A + B)^3 = 912,673

My goal is to find A and B such that this equation holds, and then add them together.

First, maybe I should figure out what the cube root of 912,673 is. If I can find the cube root, that should give me the two-digit number AB. Then I can just break it down into A and B. Let me try that.

So, let me calculate the cube root of 912,673. Hmm, cube roots can be tricky without a calculator, but maybe I can estimate it. Let me think about some cubes of numbers.

I know that 10^3 is 1000, 100^3 is 1,000,000. So, 912,673 is between 10^3 and 100^3. But since it's a two-digit number, AB is between 10 and 99. Therefore, the cube root of 912,673 should be between 10 and 100. But since it's a two-digit number, it's between 10 and 99. Let me narrow it down.

Let me think about some cubes. Let me try 90^3. 90^3 is 729,000. 95^3 is... let me calculate that. 95^3. Well, 95*95 is 9025, then multiply by 95 again. Let me do that step by step. 9025 * 95. Hmm, 9025*100 is 902,500, minus 9025*5 which is 45,125. So 902,500 - 45,125 = 857,375. So 95^3 is 857,375.

But our target is 912,673, which is higher than 857,375. Let me try 96^3. Let me compute 96^3. Well, 96^2 is 9216. Then multiply by 96. 9216*96. Let me break that down. 9216*100 is 921,600 minus 9216*4 which is 36,864. So 921,600 - 36,864 = 884,736. So 96^3 is 884,736. Still less than 912,673.

Next, 97^3. Let me compute that. 97^2 is 9409. Multiply by 97. 9409*97. Let me do 9409*100 = 940,900 minus 9409*3 = 28,227. So 940,900 - 28,227 = 912,673. Wait, that's exactly the number we have! So 97^3 is 912,673.

So that means AB is 97. Therefore, A is 9 and B is 7. Therefore, A + B is 9 + 7 = 16.

Wait, let me check that again. Let me verify 97 cubed. Let me compute 97*97 first. 97*97. Well, 100*100 is 10,000, minus 3*100 + 3*100 - 9? Wait, maybe another way. 97*97 is (90 + 7)^2 = 90^2 + 2*90*7 + 7^2 = 8100 + 1260 + 49 = 8100 + 1260 is 9360 + 49 is 9409. Then 9409*97. Let me do that again.

Alternatively, maybe I can use the fact that 97^3 = (100 - 3)^3. Let me expand that.

(100 - 3)^3 = 100^3 - 3*100^2*3 + 3*100*3^2 - 3^3. Wait, no, the formula is (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3. So here, a is 100, b is 3. Therefore:

(100 - 3)^3 = 100^3 - 3*(100)^2*3 + 3*(100)*(3)^2 - (3)^3

Calculating each term:

100^3 = 1,000,000

3*(100)^2*3 = 3*10,000*3 = 90,000

3*(100)*(3)^2 = 3*100*9 = 2,700

(3)^3 = 27

So putting it all together:

1,000,000 - 90,000 + 2,700 - 27 =

First, 1,000,000 - 90,000 = 910,000

Then, 910,000 + 2,700 = 912,700

Then, 912,700 - 27 = 912,673.

Yes! That matches the given number. So 97^3 is indeed 912,673. Therefore, AB is 97, so A is 9 and B is 7. Therefore, A + B is 9 + 7 = 16.

But let me just make sure there isn't another two-digit number that could satisfy this. For example, maybe I miscalculated? Let me check 98^3. Let me compute 98^3.

Well, 98^3. Let me do (100 - 2)^3. Using the same formula:

(100 - 2)^3 = 100^3 - 3*100^2*2 + 3*100*2^2 - 2^3

Which is 1,000,000 - 3*10,000*2 + 3*100*4 - 8

Calculates to:

1,000,000 - 60,000 + 1,200 - 8 =

1,000,000 - 60,000 = 940,000

940,000 + 1,200 = 941,200

941,200 - 8 = 941,192.

Which is higher than 912,673, so 98^3 is 941,192. So that's too big. So 97 is the correct one.

What about 96^3? Earlier, I calculated 96^3 as 884,736, which is less than 912,673. So 97 is between 96 and 98, and since 97^3 is exactly the number given, that must be the correct one.

Therefore, AB is 97, so A is 9 and B is 7. Therefore, A + B is 16.

But just to be thorough, let me check if there's any other two-digit numbers that could cube to 912,673. For example, maybe a number with leading zeros? But no, AB is a two-digit integer, so A can't be zero. So A is from 1 to 9, B is from 0 to 9. So 97 is the only one.

Alternatively, maybe I can check the cube root of 912,673 numerically. Let me try to approximate.

We know that 97^3 is 912,673. Let me check with another method. Let me take the cube root of 912,673.

Since 9.7^3 is 912.673? Wait, no, that's not. Wait, 9.7^3 is (approximate). Wait, 9.7^3 is 9.7*9.7*9.7. Let me compute that.

First, 9.7*9.7 = 94.09. Then 94.09*9.7. Let me compute that.

Well, 94.09 * 10 = 940.9, minus 94.09 * 0.3 = 28.227. So 940.9 - 28.227 = 912.673.

Oh! So 9.7^3 is 912.673. But our number is 912,673. So that's 912.673 * 1000. So that would mean that the cube root of 912,673 is 9.7 * 10 = 97. Exactly! So that confirms that the cube root is 97.

Therefore, AB is 97, so A + B is 9 + 7 = 16.

Therefore, the answer is 16.

But just to make sure, let me check once again. Let me compute 97*97*97.

First, 97*97 is 9409. Then 9409*97. Let me do this multiplication step by step.

Let me write 9409 * 97 as 9409*(100 - 3) = 9409*100 - 9409*3 = 940,900 - 28,227.

Calculating 940,900 - 28,227:

Subtract 28,000 from 940,900: 940,900 - 28,000 = 912,900

Then subtract 227 more: 912,900 - 227 = 912,673.

Yes! Perfect. So that's exactly the number given.

Therefore, I can be confident that AB is 97, so A is 9 and B is 7, so A + B is 16.

I think that's solid. I don't see any mistakes in my reasoning. I checked multiple ways: calculating the cube of 97, using the binomial expansion, approximating the cube root, and verifying the multiplication step by step. All methods lead to the same conclusion.

Therefore, the answer is 16.

**Final Answer**
\boxed{16}

To solve the problem, we are given that the cube of a two-digit integer "AB" equals 912,673. Our goal is to find the sum of its digits, $ A + B $.

---

### Step 1: Understand the Structure of the Number

The two-digit number "AB" can be expressed as:

$$
10A + B
$$

We are told that:

$$
(10A + B)^3 = 912,673
$$

So, we need to find the cube root of 912,673 to determine the value of $ 10A + B $, and then extract the digits $ A $ and $ B $.

---

### Step 2: Estimate the Cube Root

We know that:

- $ 90^3 = 729,000 $
- $ 95^3 = 857,375 $
- $ 96^3 = 884,736 $
- $ 97^3 = 912,673 $

So, the cube root of 912,673 is exactly **97**.

---

### Step 3: Identify the Digits

Since the number is 97, we have:

- $ A = 9 $
- $ B = 7 $

Therefore, the sum of the digits is:

$$
A + B = 9 + 7 = 16
$$

---

### Step 4: Verification

To ensure accuracy, we can verify that:

$$
97^3 = (100 - 3)^3 = 100^3 - 3 \cdot 100^2 \cdot 3 + 3 \cdot 100 \cdot 3^2 - 3^3
$$

$$
= 1,000,000 - 90,000 + 2,700 - 27 = 912,673
$$

This confirms that the cube of 97 is indeed 912,673.

---

### Final Answer

$$
\boxed{16}
$$

[Assistant]
Explanation:
Okay, so I need to find a two-digit integer AB such that when it's cubed, the result is 912,673. Then, I have to find the sum of its digits A and B. Alright, let me think about how to approach this.

First, let me recall that AB is a two-digit number. That means it's actually 10*A + B, right? So, for example, if A was 1 and B was 2, the number would be 12. But here, we don't know A and B yet. The problem is telling me that when I cube this number, I get 912,673. So, mathematically, that would be:

(10A + B)^3 = 912,673

My goal is to find A and B such that this equation holds, and then add them together.

First, maybe I should figure out what the cube root of 912,673 is. If I can find the cube root, that should give me the two-digit number AB. Then I can just break it down into A and B. Let me try that.

So, let me calculate the cube root of 912,673. Hmm, cube roots can be tricky without a calculator, but maybe I can estimate it. Let me think about some cubes of numbers.

I know that 10^3 is 1000, 100^3 is 1,000,000. So, 912,673 is between 10^3 and 100^3. But since it's a two-digit number, AB is between 10 and 99. Therefore, the cube root of 912,673 should be between 10 and 100. But since it's a two-digit number, it's between 10 and 99. Let me narrow it down.

Let me think about some cubes. Let me try 90^3. 90^3 is 729,000. 95^3 is... let me calculate that. 95^3. Well, 95*95 is 9025, then multiply by 95 again. Let me do that step by step. 9025 * 95. Hmm, 9025*100 is 902,500, minus 9025*5 which is 45,125. So 902,500 - 45,125 = 857,375. So 95^3 is 857,375.

But our target is 912,673, which is higher than 857,375. Let me try 96^3. Let me compute 96^3. Well, 96^2 is 9216. Then multiply by 96. 9216*96. Let me break that down. 9216*100 is 921,600 minus 9216*4 which is 36,864. So 921,600 - 36,864 = 884,736. So 96^3 is 884,736. Still less than 912,673.

Next, 97^3. Let me compute that. 97^2 is 9409. Multiply by 97. 9409*97. Let me do 9409*100 = 940,900 minus 9409*3 = 28,227. So 940,900 - 28,227 = 912,673. Wait, that's exactly the number we have! So 97^3 is 912,673.

So that means AB is 97. Therefore, A is 9 and B is 7. Therefore, A + B is 9 + 7 = 16.

Wait, let me check that again. Let me verify 97 cubed. Let me compute 97*97 first. 97*97. Well, 100*100 is 10,000, minus 3*100 + 3*100 - 9? Wait, maybe another way. 97*97 is (90 + 7)^2 = 90^2 + 2*90*7 + 7^2 = 8100 + 1260 + 49 = 8100 + 1260 is 9360 + 49 is 9409. Then 9409*97. Let me do that again.

Alternatively, maybe I can use the fact that 97^3 = (100 - 3)^3. Let me expand that.

(100 - 3)^3 = 100^3 - 3*100^2*3 + 3*100*3^2 - 3^3. Wait, no, the formula is (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3. So here, a is 100, b is 3. Therefore:

(100 - 3)^3 = 100^3 - 3*(100)^2*3 + 3*(100)*(3)^2 - (3)^3

Calculating each term:

100^3 = 1,000,000

3*(100)^2*3 = 3*10,000*3 = 90,000

3*(100)*(3)^2 = 3*100*9 = 2,700

(3)^3 = 27

So putting it all together:

1,000,000 - 90,000 + 2,700 - 27 =

First, 1,000,000 - 90,000 = 910,000

Then, 910,000 + 2,700 = 912,700

Then, 912,700 - 27 = 912,673.

Yes! That matches the given number. So 97^3 is indeed 912,673. Therefore, AB is 97, so A is 9 and B is 7. Therefore, A + B is 9 + 7 = 16.

But let me just make sure there isn't another two-digit number that could satisfy this. For example, maybe I miscalculated? Let me check 98^3. Let me compute 98^3.

Well, 98^3. Let me do (100 - 2)^3. Using the same formula:

(100 - 2)^3 = 100^3 - 3*100^2*2 + 3*100*2^2 - 2^3

Which is 1,000,000 - 3*10,000*2 + 3*100*4 - 8

Calculates to:

1,000,000 - 60,000 + 1,200 - 8 =

1,000,000 - 60,000 = 940,000

940,000 + 1,200 = 941,200

941,200 - 8 = 941,192.

Which is higher than 912,673, so 98^3 is 941,192. So that's too big. So 97 is the correct one.

What about 96^3? Earlier, I calculated 96^3 as 884,736, which is less than 912,673. So 97 is between 96 and 98, and since 97^3 is exactly the number given, that must be the correct one.

Therefore, AB is 97, so A is 9 and B is 7. Therefore, A + B is 16.

But just to be thorough, let me check if there's any other two-digit numbers that could cube to 912,673. For example, maybe a number with leading zeros? But no, AB is a two-digit integer, so A can't be zero. So A is from 1 to 9, B is from 0 to 9. So 97 is the only one.

Alternatively, maybe I can check the cube root of 912,673 numerically. Let me try to approximate.

We know that 97^3 is 912,673. Let me check with another method. Let me take the cube root of 912,673.

Since 9.7^3 is 912.673? Wait, no, that's not. Wait, 9.7^3 is (approximate). Wait, 9.7^3 is 9.7*9.7*9.7. Let me compute that.

First, 9.7*9.7 = 94.09. Then 94.09*9.7. Let me compute that.

Well, 94.09 * 10 = 940.9, minus 94.09 * 0.3 = 28.227. So 940.9 - 28.227 = 912.673.

Oh! So 9.7^3 is 912.673. But our number is 912,673. So that's 912.673 * 1000. So that would mean that the cube root of 912,673 is 9.7 * 10 = 97. Exactly! So that confirms that the cube root is 97.

Therefore, AB is 97, so A + B is 9 + 7 = 16.

Therefore, the answer is 16.

But just to make sure, let me check once again. Let me compute 97*97*97.

First, 97*97 is 9409. Then 9409*97. Let me do this multiplication step by step.

Let me write 9409 * 97 as 9409*(100 - 3) = 9409*100 - 9409*3 = 940,900 - 28,227.

Calculating 940,900 - 28,227:

Subtract 28,000 from 940,900: 940,900 - 28,000 = 912,900

Then subtract 227 more: 912,900 - 227 = 912,673.

Yes! Perfect. So that's exactly the number given.

Therefore, I can be confident that AB is 97, so A is 9 and B is 7, so A + B is 16.

I think that's solid. I don't see any mistakes in my reasoning. I checked multiple ways: calculating the cube of 97, using the binomial expansion, approximating the cube root, and verifying the multiplication step by step. All methods lead to the same conclusion.

Therefore, the answer is 16.

**Final Answer**
\boxed{16}

To solve the problem, we are given that the cube of a two-digit integer "AB" equals 912,673. Our goal is to find the sum of its digits, $ A + B $.

---

### Step 1: Understand the Structure of the Number

The two-digit number "AB" can be expressed as:

$$
10A + B
$$

We are told that:

$$
(10A + B)^3 = 912,673
$$

So, we need to find the cube root of 912,673 to determine the value of $ 10A + B $, and then extract the digits $ A $ and $ B $.

---

### Step 2: Estimate the Cube Root

We know that:

- $ 90^3 = 729,000 $
- $ 95^3 = 857,375 $
- $ 96^3 = 884,736 $
- $ 97^3 = 912,673 $

So, the cube root of 912,673 is exactly **97**.

---

### Step 3: Identify the Digits

Since the number is 97, we have:

- $ A = 9 $
- $ B = 7 $

Therefore, the sum of the digits is:

$$
A + B = 9 + 7 = 16
$$

---

### Step 4: Verification

To ensure accuracy, we can verify that:

$$
97^3 = (100 - 3)^3 = 100^3 - 3 \cdot 100^2 \cdot 3 + 3 \cdot 100 \cdot 3^2 - 3^3
$$

$$
= 1,000,000 - 90,000 + 2,700 - 27 = 912,673
$$

This confirms that the cube of 97 is indeed 912,673.

---

### Final Answer

$$
\boxed{16}
$$

Solution:
Since $90^3=729,\!000$, $\text{AB}$ is greater than 90.  Therefore, $\text{A}=9$.  Since the ones digit of $\text{AB}^3$ is 3, $\text{AB}$ must be odd.  The ones digit of $\text{AB}^3$ is the same as the ones digit of $\text{B}^3$, so we look at the ones digits of the cubes of the odd digits. \[
\begin{array}{c}
\text{The ones digit of }1^3 \text{ is } 1. \\ \text{The ones digit of }3^3 \text{ is } 7. \\ \text{The ones digit of }5^3 \text{ is } 5. \\ \text{The ones digit of }7^3 \text{ is } 3. \\ \text{The ones digit of }9^3 \text{ is } 9.
\end{array}
\] Only $7^3$ has a ones digit of 3, so $\text{B}=7$.  Therefore, $\text{A}+\text{B}=9+7=\boxed{16}$.


Explanation:
Okay, so I need to figure out how to compute this expression: 5^5 divided by 5^4 minus 5^3 plus 5^2 times 5^1. Let me write that down again to make sure I have it right. It's 5^5 ÷ 5^4 - 5^3 + 5^2 · 5^1. Alright, let me break this down step by step.

First, I remember that when you divide exponents with the same base, you subtract the exponents. So 5^5 divided by 5^4 should be 5^(5-4), which is 5^1. Let me check that. Yeah, because 5^5 is 5*5*5*5*5 and 5^4 is 5*5*5*5, so when you divide them, you're left with one 5. So that part simplifies to 5.

Okay, so now the expression becomes 5 - 5^3 + 5^2 · 5^1. Let me handle the next part. The next term is -5^3. Let me compute that. 5^3 is 5*5*5, which is 125. So that term is -125.

Now the expression is 5 - 125 + ... Then there's the last part: 5^2 multiplied by 5^1. Again, when multiplying exponents with the same base, you add the exponents. So 5^(2+1) = 5^3. Let me verify that. 5^2 is 25 and 5^1 is 5, so 25*5 is 125. Yep, that's 5^3, which is 125.

So substituting that back into the expression, we have 5 - 125 + 125. Let me write that out: 5 - 125 + 125. Hmm, let me compute this step by step.

Starting with 5 - 125. That would be... 5 minus 125. Let me think. 125 - 5 is 120, so 5 - 125 is -120. Then, adding 125 to that. So -120 + 125. That should be 5. Wait, is that right? Let me check again.

Starting with 5 - 125 + 125. If I rearrange the terms, it's 5 + (-125 + 125). The -125 and +125 cancel each other out, leaving just 5. So the final result is 5? That seems too simple. Let me go through each step again to make sure I didn't make a mistake.

Original expression: 5^5 ÷ 5^4 - 5^3 + 5^2 · 5^1.

First term: 5^5 / 5^4. As per exponent rules, subtract exponents: 5^(5-4) = 5^1 = 5. Correct.

Second term: -5^3. So that's - (5^3) = -125. Correct.

Third term: 5^2 * 5^1. Add exponents: 5^(2+1) = 5^3 = 125. Correct.

So substituting back: 5 - 125 + 125.

Now, combining the terms: 5 + (-125 + 125) = 5 + 0 = 5.

Wait, so the answer is 5? That seems straightforward, but let me check by calculating each term numerically without using exponent rules, just to confirm.

Calculating each term:

First term: 5^5 is 3125. Divided by 5^4, which is 625. So 3125 / 625. Let me compute that. 625 * 5 is 3125, so 3125 / 625 = 5. Correct.

Second term: -5^3. 5^3 is 125, so -125. Correct.

Third term: 5^2 is 25, 5^1 is 5. 25 * 5 is 125. Correct.

So putting it all together: 5 - 125 + 125.

Calculating left to right: 5 - 125 is -120. Then -120 + 125 is 5. Yep, same result.

So the answer is 5. Wait, but let me check if there's any order of operations I might have messed up. The original expression is 5^5 ÷ 5^4 - 5^3 + 5^2 · 5^1.

Order of operations: exponents, then multiplication/division from left to right, then addition/subtraction from left to right.

So first, compute all exponents:

5^5 = 3125, 5^4 = 625, 5^3 = 125, 5^2 = 25, 5^1 = 5.

So substituting back:

3125 ÷ 625 - 125 + 25 * 5.

Now compute division and multiplication first:

3125 ÷ 625 = 5.

25 * 5 = 125.

So now the expression is 5 - 125 + 125.

Then, left to right: 5 - 125 = -120; -120 + 125 = 5.

Same result. So seems like 5 is the correct answer.

But just to make sure, let me think if there's any alternative interpretation. For example, could the original expression be interpreted differently? Like, maybe the subtraction and addition have different precedence? But no, according to order of operations, after exponents, we do multiplication and division left to right, then addition and subtraction left to right.

So the original expression is:

(5^5 ÷ 5^4) - 5^3 + (5^2 · 5^1).

So that's (5) - 125 + 125.

Alternatively, if someone misapplies the order of operations, maybe they might do something else, but I don't think so.

Alternatively, could there be a mistake in the exponent rules? Let me check again:

For division: 5^5 / 5^4 = 5^(5-4) = 5^1. Correct.

For multiplication: 5^2 * 5^1 = 5^(2+1) = 5^3. Correct.

So that's right.

Alternatively, if someone thought that 5^5 ÷ 5^4 - 5^3 + 5^2 · 5^1 is grouped differently, but unless there are parentheses, which there aren't, it's left to right.

So, yeah, I think 5 is the correct answer.

But just to be absolutely thorough, let me compute all the numbers step by step without exponents:

Original expression:

5^5 ÷ 5^4 - 5^3 + 5^2 · 5^1.

Compute each term:

First term: 5^5 ÷ 5^4.

Compute 5^5: 5*5=25, 25*5=125, 125*5=625, 625*5=3125.

Compute 5^4: 5*5=25, 25*5=125, 125*5=625.

Divide 3125 by 625: 3125 / 625. Let me divide numerator and denominator by 25: 3125 ÷25=125, 625 ÷25=25. So 125 /25=5. Correct.

Second term: -5^3.

Compute 5^3: 5*5=25, 25*5=125. So -125.

Third term: 5^2 * 5^1.

Compute 5^2: 25. 5^1: 5. Multiply them: 25*5=125.

So putting it all together: 5 - 125 + 125.

Compute 5 - 125: 5 - 125 = -120.

Then -120 + 125: 5.

Yep, same result.

Therefore, I think the answer is 5.

But just to make sure, let me think if there's any chance that the original problem was written differently. For example, if there were parentheses that changed the order, but the original problem is written as:

5^5 ÷ 5^4 - 5^3 + 5^2 · 5^1.

So no parentheses, so order of operations is as we did.

Alternatively, if someone thought that the subtraction applies to the entire 5^3 + ... but no, the expression is 5^5 ÷ 5^4 - 5^3 + 5^2 · 5^1. So the subtraction is only of 5^3, and then addition of the product.

Alternatively, if someone misreads the expression as 5^5 ÷ (5^4 - 5^3) + 5^2 · 5^1, but that would require parentheses, which aren't there.

So, no, I think the answer is definitely 5.

Therefore, I can confidently say the integer result is 5.

**Final Answer**
\boxed{5}

To evaluate the expression:

$$
5^5 \div 5^4 - 5^3 + 5^2 \cdot 5^1
$$

we proceed step by step, applying the rules of exponents and the order of operations (PEMDAS/BODMAS).

---

### Step 1: Simplify the Division

We start with the division of exponents with the same base:

$$
5^5 \div 5^4 = 5^{5-4} = 5^1 = 5
$$

---

### Step 2: Simplify the Multiplication

Next, we simplify the multiplication of exponents with the same base:

$$
5^2 \cdot 5^1 = 5^{2+1} = 5^3 = 125
$$

---

### Step 3: Substitute Back Into the Expression

Now substitute the simplified values back into the original expression:

$$
5 - 5^3 + 5^2 \cdot 5^1 = 5 - 125 + 125
$$

---

### Step 4: Perform the Arithmetic

Now compute the expression from left to right:

$$
5 - 125 = -120
$$
$$
-120 + 125 = 5
$$

---

### Final Answer

$$
\boxed{5}
$$

[Assistant]
Explanation:
Okay, so I need to figure out how to compute this expression: 5^5 divided by 5^4 minus 5^3 plus 5^2 times 5^1. Let me write that down again to make sure I have it right. It's 5^5 ÷ 5^4 - 5^3 + 5^2 · 5^1. Alright, let me break this down step by step.

First, I remember that when you divide exponents with the same base, you subtract the exponents. So 5^5 divided by 5^4 should be 5^(5-4), which is 5^1. Let me check that. Yeah, because 5^5 is 5*5*5*5*5 and 5^4 is 5*5*5*5, so when you divide them, you're left with one 5. So that part simplifies to 5.

Okay, so now the expression becomes 5 - 5^3 + 5^2 · 5^1. Let me handle the next part. The next term is -5^3. Let me compute that. 5^3 is 5*5*5, which is 125. So that term is -125.

Now the expression is 5 - 125 + ... Then there's the last part: 5^2 multiplied by 5^1. Again, when multiplying exponents with the same base, you add the exponents. So 5^(2+1) = 5^3. Let me verify that. 5^2 is 25 and 5^1 is 5, so 25*5 is 125. Yep, that's 5^3, which is 125.

So substituting that back into the expression, we have 5 - 125 + 125. Let me write that out: 5 - 125 + 125. Hmm, let me compute this step by step.

Starting with 5 - 125. That would be... 5 minus 125. Let me think. 125 - 5 is 120, so 5 - 125 is -120. Then, adding 125 to that. So -120 + 125. That should be 5. Wait, is that right? Let me check again.

Starting with 5 - 125 + 125. If I rearrange the terms, it's 5 + (-125 + 125). The -125 and +125 cancel each other out, leaving just 5. So the final result is 5? That seems too simple. Let me go through each step again to make sure I didn't make a mistake.

Original expression: 5^5 ÷ 5^4 - 5^3 + 5^2 · 5^1.

First term: 5^5 / 5^4. As per exponent rules, subtract exponents: 5^(5-4) = 5^1 = 5. Correct.

Second term: -5^3. So that's - (5^3) = -125. Correct.

Third term: 5^2 * 5^1. Add exponents: 5^(2+1) = 5^3 = 125. Correct.

So substituting back: 5 - 125 + 125.

Now, combining the terms: 5 + (-125 + 125) = 5 + 0 = 5.

Wait, so the answer is 5? That seems straightforward, but let me check by calculating each term numerically without using exponent rules, just to confirm.

Calculating each term:

First term: 5^5 is 3125. Divided by 5^4, which is 625. So 3125 / 625. Let me compute that. 625 * 5 is 3125, so 3125 / 625 = 5. Correct.

Second term: -5^3. 5^3 is 125, so -125. Correct.

Third term: 5^2 is 25, 5^1 is 5. 25 * 5 is 125. Correct.

So putting it all together: 5 - 125 + 125.

Calculating left to right: 5 - 125 is -120. Then -120 + 125 is 5. Yep, same result.

So the answer is 5. Wait, but let me check if there's any order of operations I might have messed up. The original expression is 5^5 ÷ 5^4 - 5^3 + 5^2 · 5^1.

Order of operations: exponents, then multiplication/division from left to right, then addition/subtraction from left to right.

So first, compute all exponents:

5^5 = 3125, 5^4 = 625, 5^3 = 125, 5^2 = 25, 5^1 = 5.

So substituting back:

3125 ÷ 625 - 125 + 25 * 5.

Now compute division and multiplication first:

3125 ÷ 625 = 5.

25 * 5 = 125.

So now the expression is 5 - 125 + 125.

Then, left to right: 5 - 125 = -120; -120 + 125 = 5.

Same result. So seems like 5 is the correct answer.

But just to make sure, let me think if there's any alternative interpretation. For example, could the original expression be interpreted differently? Like, maybe the subtraction and addition have different precedence? But no, according to order of operations, after exponents, we do multiplication and division left to right, then addition and subtraction left to right.

So the original expression is:

(5^5 ÷ 5^4) - 5^3 + (5^2 · 5^1).

So that's (5) - 125 + 125.

Alternatively, if someone misapplies the order of operations, maybe they might do something else, but I don't think so.

Alternatively, could there be a mistake in the exponent rules? Let me check again:

For division: 5^5 / 5^4 = 5^(5-4) = 5^1. Correct.

For multiplication: 5^2 * 5^1 = 5^(2+1) = 5^3. Correct.

So that's right.

Alternatively, if someone thought that 5^5 ÷ 5^4 - 5^3 + 5^2 · 5^1 is grouped differently, but unless there are parentheses, which there aren't, it's left to right.

So, yeah, I think 5 is the correct answer.

But just to be absolutely thorough, let me compute all the numbers step by step without exponents:

Original expression:

5^5 ÷ 5^4 - 5^3 + 5^2 · 5^1.

Compute each term:

First term: 5^5 ÷ 5^4.

Compute 5^5: 5*5=25, 25*5=125, 125*5=625, 625*5=3125.

Compute 5^4: 5*5=25, 25*5=125, 125*5=625.

Divide 3125 by 625: 3125 / 625. Let me divide numerator and denominator by 25: 3125 ÷25=125, 625 ÷25=25. So 125 /25=5. Correct.

Second term: -5^3.

Compute 5^3: 5*5=25, 25*5=125. So -125.

Third term: 5^2 * 5^1.

Compute 5^2: 25. 5^1: 5. Multiply them: 25*5=125.

So putting it all together: 5 - 125 + 125.

Compute 5 - 125: 5 - 125 = -120.

Then -120 + 125: 5.

Yep, same result.

Therefore, I think the answer is 5.

But just to make sure, let me think if there's any chance that the original problem was written differently. For example, if there were parentheses that changed the order, but the original problem is written as:

5^5 ÷ 5^4 - 5^3 + 5^2 · 5^1.

So no parentheses, so order of operations is as we did.

Alternatively, if someone thought that the subtraction applies to the entire 5^3 + ... but no, the expression is 5^5 ÷ 5^4 - 5^3 + 5^2 · 5^1. So the subtraction is only of 5^3, and then addition of the product.

Alternatively, if someone misreads the expression as 5^5 ÷ (5^4 - 5^3) + 5^2 · 5^1, but that would require parentheses, which aren't there.

So, no, I think the answer is definitely 5.

Therefore, I can confidently say the integer result is 5.

**Final Answer**
\boxed{5}

To evaluate the expression:

$$
5^5 \div 5^4 - 5^3 + 5^2 \cdot 5^1
$$

we proceed step by step, applying the rules of exponents and the order of operations (PEMDAS/BODMAS).

---

### Step 1: Simplify the Division

We start with the division of exponents with the same base:

$$
5^5 \div 5^4 = 5^{5-4} = 5^1 = 5
$$

---

### Step 2: Simplify the Multiplication

Next, we simplify the multiplication of exponents with the same base:

$$
5^2 \cdot 5^1 = 5^{2+1} = 5^3 = 125
$$

---

### Step 3: Substitute Back Into the Expression

Now substitute the simplified values back into the original expression:

$$
5 - 5^3 + 5^2 \cdot 5^1 = 5 - 125 + 125
$$

---

### Step 4: Perform the Arithmetic

Now compute the expression from left to right:

$$
5 - 125 = -120
$$
$$
-120 + 125 = 5
$$

---

### Final Answer

$$
\boxed{5}
$$

Solution:
Recall that $a^m\div a^n=a^{m-n}$ for positive integers $m>n$ and $a^m\cdot a^n=a^{m+n}$. Now we can write $5^5\div5^4-5^3+5^2\cdot5^1$ as $5^1-5^3+5^3$. Using the definition of subtraction and the associative property of addition, we get \begin{align*}
5^1-5^3+5^3&=5^1+-5^3+5^3\\
&=5^1+(-5^3+5^3)\\
&=5^1+0\\
&=\boxed{5}.
\end{align*}


Explanation:
Okay, so I need to find the midpoint between two points, (-5, 5) and (3, 7). Hmm, let me remember how to do this. I think there's a formula for the midpoint of a line segment. Let me recall... I think it's something like averaging the x-coordinates and the y-coordinates of the two endpoints. Yeah, that sounds right.

So, if I have two points, let's call them (x1, y1) and (x2, y2), then the midpoint (let's call it M) should be ((x1 + x2)/2, (y1 + y2)/2). Is that correct? Let me check with a simple example. Suppose I have points (0,0) and (2,2). The midpoint should be (1,1). Using the formula: ((0+2)/2, (0+2)/2) = (1,1). Yep, that works. So the formula seems right.

Alright, so applying this to the given points. Let me write down the coordinates:

First point: (-5, 5). So x1 is -5, y1 is 5.

Second point: (3, 7). So x2 is 3, y2 is 7.

Now, let me compute the x-coordinate of the midpoint. That would be (x1 + x2)/2. Plugging in the numbers: (-5 + 3)/2. Let me calculate that. -5 + 3 is... -2. Then divide by 2: -2/2 = -1. Okay, so the x-coordinate is -1.

Now the y-coordinate: (y1 + y2)/2. That's (5 + 7)/2. 5 + 7 is 12. Divided by 2 is 6. So the y-coordinate is 6.

Therefore, the midpoint should be (-1, 6). Let me just verify that. If I imagine the two points, (-5,5) and (3,7), the midpoint should be halfway between them. Let me check the horizontal distance first. From x = -5 to x = 3. The difference is 3 - (-5) = 8. Half of that is 4. So starting from -5, adding 4 would get me to -5 + 4 = -1. That matches the x-coordinate we found.

For the vertical distance, from y = 5 to y = 7. The difference is 7 - 5 = 2. Half of that is 1. Starting from 5, adding 1 gives 6. Which matches the y-coordinate. So that seems to check out.

Wait, let me think again. Another way to check is to see if the midpoint is equidistant from both points. Let me calculate the distance from (-1,6) to (-5,5) and to (3,7).

First distance: sqrt[(-1 - (-5))² + (6 - 5)²] = sqrt[(4)² + (1)²] = sqrt[16 + 1] = sqrt[17].

Second distance: sqrt[(3 - (-1))² + (7 - 6)²] = sqrt[(4)² + (1)²] = sqrt[16 + 1] = sqrt[17].

Same distance, so that's good. So the midpoint is indeed equidistant from both endpoints, which confirms that (-1, 6) is correct.

Alternatively, I could also think of the midpoint as the average of the coordinates. So, for the x-coordinate, average of -5 and 3. Let me compute that again. (-5 + 3)/2 = (-2)/2 = -1. Yep. For the y-coordinate, average of 5 and 7. (5 + 7)/2 = 12/2 = 6. Yep.

I think that's solid. I don't see any mistakes here. Maybe I can visualize it on a coordinate plane. Let me sketch mentally. The first point is in the second quadrant, since x is negative and y is positive. The second point is in the first quadrant, since both coordinates are positive. The midpoint should be somewhere between them.

Starting at (-5,5), moving towards (3,7). The x-coordinate goes from -5 to 3, which is moving right 8 units. The y-coordinate goes from 5 to 7, which is moving up 2 units. The midpoint would be halfway along that path. So halfway in x is -5 + 4 = -1, and halfway in y is 5 + 1 = 6. Yep, that's (-1,6).

Another way to think about it: if I consider vectors. The vector from (-5,5) to (3,7) is (3 - (-5), 7 - 5) = (8, 2). Half of that vector is (4, 1). Adding that to the starting point (-5,5) gives (-5 + 4, 5 + 1) = (-1, 6). Same result.

So all methods lead to (-1, 6). Therefore, I can be confident that the midpoint is (-1, 6).

Just to recap, the formula is straightforward: average the x's and average the y's. It's important to remember that the midpoint is not just any point between the two, but specifically the one that's exactly halfway. The formula ensures that.

I don't think I made any calculation errors here. Let me check once more:

x1 + x2 = -5 + 3 = -2. Divided by 2 is -1. Correct.

y1 + y2 = 5 + 7 = 12. Divided by 2 is 6. Correct.

Yep, all steps are correct. So the answer is (-1, 6).

**Final Answer**
The midpoint of the line segment is \boxed{(-1, 6)}.

To find the midpoint of a line segment between two points, we use the **midpoint formula**:

$$
\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
$$

---

### Step 1: Identify the coordinates of the endpoints

We are given the two endpoints of the line segment:

- Point A: $(-5, 5)$
- Point B: $(3, 7)$

So, we have:
- $x_1 = -5$, $y_1 = 5$
- $x_2 = 3$, $y_2 = 7$

---

### Step 2: Apply the midpoint formula

#### Compute the x-coordinate of the midpoint:

$$
\frac{x_1 + x_2}{2} = \frac{-5 + 3}{2} = \frac{-2}{2} = -1
$$

#### Compute the y-coordinate of the midpoint:

$$
\frac{y_1 + y_2}{2} = \frac{5 + 7}{2} = \frac{12}{2} = 6
$$

---

### Step 3: Write the final answer

The midpoint of the line segment is:

$$
\boxed{(-1, 6)}
$$

[Assistant]
Explanation:
Okay, so I need to find the midpoint between two points, (-5, 5) and (3, 7). Hmm, let me remember how to do this. I think there's a formula for the midpoint of a line segment. Let me recall... I think it's something like averaging the x-coordinates and the y-coordinates of the two endpoints. Yeah, that sounds right.

So, if I have two points, let's call them (x1, y1) and (x2, y2), then the midpoint (let's call it M) should be ((x1 + x2)/2, (y1 + y2)/2). Is that correct? Let me check with a simple example. Suppose I have points (0,0) and (2,2). The midpoint should be (1,1). Using the formula: ((0+2)/2, (0+2)/2) = (1,1). Yep, that works. So the formula seems right.

Alright, so applying this to the given points. Let me write down the coordinates:

First point: (-5, 5). So x1 is -5, y1 is 5.

Second point: (3, 7). So x2 is 3, y2 is 7.

Now, let me compute the x-coordinate of the midpoint. That would be (x1 + x2)/2. Plugging in the numbers: (-5 + 3)/2. Let me calculate that. -5 + 3 is... -2. Then divide by 2: -2/2 = -1. Okay, so the x-coordinate is -1.

Now the y-coordinate: (y1 + y2)/2. That's (5 + 7)/2. 5 + 7 is 12. Divided by 2 is 6. So the y-coordinate is 6.

Therefore, the midpoint should be (-1, 6). Let me just verify that. If I imagine the two points, (-5,5) and (3,7), the midpoint should be halfway between them. Let me check the horizontal distance first. From x = -5 to x = 3. The difference is 3 - (-5) = 8. Half of that is 4. So starting from -5, adding 4 would get me to -5 + 4 = -1. That matches the x-coordinate we found.

For the vertical distance, from y = 5 to y = 7. The difference is 7 - 5 = 2. Half of that is 1. Starting from 5, adding 1 gives 6. Which matches the y-coordinate. So that seems to check out.

Wait, let me think again. Another way to check is to see if the midpoint is equidistant from both points. Let me calculate the distance from (-1,6) to (-5,5) and to (3,7).

First distance: sqrt[(-1 - (-5))² + (6 - 5)²] = sqrt[(4)² + (1)²] = sqrt[16 + 1] = sqrt[17].

Second distance: sqrt[(3 - (-1))² + (7 - 6)²] = sqrt[(4)² + (1)²] = sqrt[16 + 1] = sqrt[17].

Same distance, so that's good. So the midpoint is indeed equidistant from both endpoints, which confirms that (-1, 6) is correct.

Alternatively, I could also think of the midpoint as the average of the coordinates. So, for the x-coordinate, average of -5 and 3. Let me compute that again. (-5 + 3)/2 = (-2)/2 = -1. Yep. For the y-coordinate, average of 5 and 7. (5 + 7)/2 = 12/2 = 6. Yep.

I think that's solid. I don't see any mistakes here. Maybe I can visualize it on a coordinate plane. Let me sketch mentally. The first point is in the second quadrant, since x is negative and y is positive. The second point is in the first quadrant, since both coordinates are positive. The midpoint should be somewhere between them.

Starting at (-5,5), moving towards (3,7). The x-coordinate goes from -5 to 3, which is moving right 8 units. The y-coordinate goes from 5 to 7, which is moving up 2 units. The midpoint would be halfway along that path. So halfway in x is -5 + 4 = -1, and halfway in y is 5 + 1 = 6. Yep, that's (-1,6).

Another way to think about it: if I consider vectors. The vector from (-5,5) to (3,7) is (3 - (-5), 7 - 5) = (8, 2). Half of that vector is (4, 1). Adding that to the starting point (-5,5) gives (-5 + 4, 5 + 1) = (-1, 6). Same result.

So all methods lead to (-1, 6). Therefore, I can be confident that the midpoint is (-1, 6).

Just to recap, the formula is straightforward: average the x's and average the y's. It's important to remember that the midpoint is not just any point between the two, but specifically the one that's exactly halfway. The formula ensures that.

I don't think I made any calculation errors here. Let me check once more:

x1 + x2 = -5 + 3 = -2. Divided by 2 is -1. Correct.

y1 + y2 = 5 + 7 = 12. Divided by 2 is 6. Correct.

Yep, all steps are correct. So the answer is (-1, 6).

**Final Answer**
The midpoint of the line segment is \boxed{(-1, 6)}.

To find the midpoint of a line segment between two points, we use the **midpoint formula**:

$$
\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
$$

---

### Step 1: Identify the coordinates of the endpoints

We are given the two endpoints of the line segment:

- Point A: $(-5, 5)$
- Point B: $(3, 7)$

So, we have:
- $x_1 = -5$, $y_1 = 5$
- $x_2 = 3$, $y_2 = 7$

---

### Step 2: Apply the midpoint formula

#### Compute the x-coordinate of the midpoint:

$$
\frac{x_1 + x_2}{2} = \frac{-5 + 3}{2} = \frac{-2}{2} = -1
$$

#### Compute the y-coordinate of the midpoint:

$$
\frac{y_1 + y_2}{2} = \frac{5 + 7}{2} = \frac{12}{2} = 6
$$

---

### Step 3: Write the final answer

The midpoint of the line segment is:

$$
\boxed{(-1, 6)}
$$

Solution: Applying the midpoint formula gives us $$\left(\frac{-5+3}{2},\frac{5+7}{2}\right)=\boxed{(-1,6)}.$$


Explanation:
Okay, so I need to find the smallest possible angle between vectors a and b, given their magnitudes and the cross product. Let me recall what I know about cross products and angles between vectors.

First, the magnitude of the cross product of two vectors a and b is given by the formula:

||a × b|| = ||a|| ||b|| sinθ

where θ is the angle between them. So, if I can find sinθ, I can then find θ. But wait, the problem gives me the cross product vector itself, not just its magnitude. Let me check the given cross product:

a × b = (3, 2, 6). So, the magnitude of this vector is sqrt(3² + 2² + 6²). Let me compute that.

Calculating the magnitude:

sqrt(9 + 4 + 36) = sqrt(49) = 7. Oh, interesting! The magnitude of the cross product is 7.

Given that ||a|| is 2 and ||b|| is 7, plugging into the formula:

||a × b|| = ||a|| ||b|| sinθ => 7 = 2 * 7 * sinθ

Simplify that:

7 = 14 sinθ => sinθ = 7 / 14 = 1/2

So sinθ = 1/2. Then θ is arcsin(1/2). The angles that satisfy this are 30 degrees and 150 degrees, since sine is positive in the first and second quadrants. But the question asks for the smallest possible angle between a and b. So between 30 degrees and 150 degrees, the smaller one is 30 degrees. Therefore, the smallest possible angle is 30 degrees.

Wait, but hold on. Is there any possibility that the cross product's magnitude could be different? Wait, no, because the cross product's magnitude is determined by the given vector. The problem says that a × b is equal to that specific vector, so its magnitude is fixed. Therefore, sinθ is fixed as 1/2, so θ is either 30° or 150°, and the smallest is 30°.

But let me double-check my calculations. Let me verify the magnitude of the cross product again. The cross product vector is (3, 2, 6). So, 3 squared is 9, 2 squared is 4, 6 squared is 36. Adding them up: 9 + 4 is 13, plus 36 is 49. Square root of 49 is 7. Yes, that's correct.

Then, ||a × b|| = 7. Then, using the formula:

7 = ||a|| ||b|| sinθ => 7 = 2 * 7 * sinθ => sinθ = 7 / 14 = 1/2. Correct.

So θ = arcsin(1/2). The principal value of arcsin(1/2) is 30°, but since sine is positive in both first and second quadrants, the angle could be 30° or 150°. However, the angle between two vectors is always taken as the smallest angle between them, which is between 0° and 180°, so the smallest possible angle is 30°, and 150° is the obtuse angle. Therefore, the answer should be 30 degrees.

But wait, the problem says "the smallest possible angle between a and b". Is there a scenario where the angle could be smaller? For example, if there are multiple possible angles? But given that sinθ is fixed, θ is uniquely determined up to the angle's supplement. However, since angles between vectors are defined as the smallest one between them, so θ is between 0° and 180°, so the smallest possible angle is 30°, and the other possibility is 150°, which is larger. Therefore, 30° is the answer.

But let me think again. Is there a possibility that the cross product's magnitude could be different? For example, if the cross product vector is given, does that affect anything else? For example, could there be different vectors a and b with the same magnitudes and cross product vector but different angles? Wait, no. Because the magnitude of the cross product is fixed by the formula ||a × b|| = ||a|| ||b|| sinθ. So if the cross product's magnitude is fixed, then sinθ is fixed. Therefore, θ is fixed up to the angle's supplement. But since the angle between vectors is defined as the smallest one, it's 30°.

Alternatively, maybe there's a mistake in assuming that the cross product's magnitude is 7? Let me check again. The cross product vector is (3, 2, 6). So, the magnitude is sqrt(3² + 2² + 6²) = sqrt(9 + 4 + 36) = sqrt(49) = 7. Yes, that's correct.

So, then sinθ = 1/2. Therefore, θ is 30° or 150°, and the smallest angle is 30°. Therefore, the answer is 30 degrees.

But let me check if there's another way to approach this problem. For example, using the dot product. The dot product formula is:

a · b = ||a|| ||b|| cosθ

But we don't have the dot product given. However, if we can find the dot product, we could find cosθ. But since we don't have the dot product, maybe we can relate it with the cross product?

Alternatively, we know that for any two vectors, the magnitude of the cross product squared plus the square of the dot product equals the product of the magnitudes squared times (sin²θ + cos²θ) = ||a||² ||b||².

So, ||a × b||² + (a · b)² = ||a||² ||b||².

Let me check that.

Yes, because:

||a × b||² + (a · b)² = (||a|| ||b|| sinθ)^2 + (||a|| ||b|| cosθ)^2 = ||a||² ||b||² (sin²θ + cos²θ) = ||a||² ||b||².

So, that's an identity.

So, if I compute ||a × b||², which is 7² = 49, and ||a||² ||b||² is 2² * 7² = 4 * 49 = 196. Therefore, (a · b)^2 = 196 - 49 = 147. Therefore, a · b = ±sqrt(147).

But sqrt(147) is sqrt(49*3) = 7*sqrt(3). Therefore, a · b = ±7√3.

Then, since a · b = ||a|| ||b|| cosθ, so:

±7√3 = 2 * 7 * cosθ => ±√3 = 2 cosθ => cosθ = ±√3 / 2.

So, cosθ is either √3 / 2 or -√3 / 2.

Therefore, θ is either 30° or 150°, since cos(30°) = √3 / 2 and cos(150°) = -√3 / 2.

Therefore, again, the angle between the vectors is either 30° or 150°, so the smallest possible angle is 30°.

Therefore, this confirms the previous result.

So, both methods lead to the same conclusion. Therefore, the answer is 30 degrees.

But let me think again if there's any other consideration. For example, could there be a case where the cross product vector is given, but the angle is different? For example, if the cross product vector is fixed, does that impose any constraints on the angle?

Wait, the cross product vector's direction is perpendicular to both a and b. So, the direction of a × b is determined by the right-hand rule. However, the problem doesn't specify any particular direction, just the vector. But since we are only given the magnitude of the cross product, which we already used, and the direction is not affecting the magnitude. Therefore, regardless of the direction of the cross product, the magnitude is fixed, so sinθ is fixed. Therefore, θ is fixed as 30° or 150°, and the smallest angle is 30°.

Therefore, I think the answer is 30 degrees.

But let me check if there's a possible mistake in assuming that the angle between vectors is always between 0° and 180°. Yes, by definition, the angle between two vectors is the smallest angle between them when they are placed tail to tail, so it's always between 0° and 180°, so 30° is indeed the smallest possible angle.

Alternatively, if someone thought that the angle could be negative or something, but no, angles between vectors are always non-negative and less than or equal to 180°.

Therefore, I think I can be confident that the answer is 30 degrees.

**Final Answer**
The smallest possible angle between vectors $\mathbf{a}$ and $\mathbf{b}$ is $\boxed{30}$ degrees.

To determine the smallest possible angle between two vectors $\mathbf{a}$ and $\mathbf{b}$, we are given the following:

- $\|\mathbf{a}\| = 2$
- $\|\mathbf{b}\| = 7$
- $\mathbf{a} \times \mathbf{b} = \begin{pmatrix} 3 \\ 2 \\ 6 \end{pmatrix}$

---

### Step 1: Use the Cross Product Magnitude Formula

The magnitude of the cross product of two vectors is given by:

$$
\|\mathbf{a} \times \mathbf{b}\| = \|\mathbf{a}\| \|\mathbf{b}\| \sin\theta
$$

We are given the vector $\mathbf{a} \times \mathbf{b} = \begin{pmatrix} 3 \\ 2 \\ 6 \end{pmatrix}$, so we can compute its magnitude:

$$
\|\mathbf{a} \times \mathbf{b}\| = \sqrt{3^2 + 2^2 + 6^2} = \sqrt{9 + 4 + 36} = \sqrt{49} = 7
$$

Now plug into the formula:

$$
7 = 2 \cdot 7 \cdot \sin\theta \Rightarrow \sin\theta = \frac{7}{14} = \frac{1}{2}
$$

---

### Step 2: Determine the Angle $\theta$

We now solve for $\theta$ such that $\sin\theta = \frac{1}{2}$. The solutions in the interval $[0^\circ, 180^\circ]$ are:

$$
\theta = 30^\circ \quad \text{or} \quad \theta = 150^\circ
$$

Since the angle between two vectors is defined as the **smallest** angle between them, the smallest possible angle is:

$$
\boxed{30}
$$

---

### Step 3: Confirm with the Dot Product (Optional Verification)

We can also use the identity:

$$
\|\mathbf{a} \times \mathbf{b}\|^2 + (\mathbf{a} \cdot \mathbf{b})^2 = \|\mathbf{a}\|^2 \|\mathbf{b}\|^2
$$

We already know:

- $\|\mathbf{a} \times \mathbf{b}\| = 7 \Rightarrow \|\mathbf{a} \times \mathbf{b}\|^2 = 49$
- $\|\mathbf{a}\|^2 = 4$, $\|\mathbf{b}\|^2 = 49 \Rightarrow \|\mathbf{a}\|^2 \|\mathbf{b}\|^2 = 196$

So:

$$
(\mathbf{a} \cdot \mathbf{b})^2 = 196 - 49 = 147 \Rightarrow \mathbf{a} \cdot \mathbf{b} = \pm \sqrt{147} = \pm 7\sqrt{3}
$$

Now use the dot product formula:

$$
\mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \|\mathbf{b}\| \cos\theta = 2 \cdot 7 \cdot \cos\theta = 14 \cos\theta
$$

So:

$$
14 \cos\theta = \pm 7\sqrt{3} \Rightarrow \cos\theta = \pm \frac{\sqrt{3}}{2}
$$

This gives:

- $\cos\theta = \frac{\sqrt{3}}{2} \Rightarrow \theta = 30^\circ$
- $\cos\theta = -\frac{\sqrt{3}}{2} \Rightarrow \theta = 150^\circ$

Again, the smallest angle is:

$$
\boxed{30}
$$

---

### Final Answer

$$
\boxed{30}
$$

[Assistant]
Explanation:
Okay, so I need to find the smallest possible angle between vectors a and b, given their magnitudes and the cross product. Let me recall what I know about cross products and angles between vectors.

First, the magnitude of the cross product of two vectors a and b is given by the formula:

||a × b|| = ||a|| ||b|| sinθ

where θ is the angle between them. So, if I can find sinθ, I can then find θ. But wait, the problem gives me the cross product vector itself, not just its magnitude. Let me check the given cross product:

a × b = (3, 2, 6). So, the magnitude of this vector is sqrt(3² + 2² + 6²). Let me compute that.

Calculating the magnitude:

sqrt(9 + 4 + 36) = sqrt(49) = 7. Oh, interesting! The magnitude of the cross product is 7.

Given that ||a|| is 2 and ||b|| is 7, plugging into the formula:

||a × b|| = ||a|| ||b|| sinθ => 7 = 2 * 7 * sinθ

Simplify that:

7 = 14 sinθ => sinθ = 7 / 14 = 1/2

So sinθ = 1/2. Then θ is arcsin(1/2). The angles that satisfy this are 30 degrees and 150 degrees, since sine is positive in the first and second quadrants. But the question asks for the smallest possible angle between a and b. So between 30 degrees and 150 degrees, the smaller one is 30 degrees. Therefore, the smallest possible angle is 30 degrees.

Wait, but hold on. Is there any possibility that the cross product's magnitude could be different? Wait, no, because the cross product's magnitude is determined by the given vector. The problem says that a × b is equal to that specific vector, so its magnitude is fixed. Therefore, sinθ is fixed as 1/2, so θ is either 30° or 150°, and the smallest is 30°.

But let me double-check my calculations. Let me verify the magnitude of the cross product again. The cross product vector is (3, 2, 6). So, 3 squared is 9, 2 squared is 4, 6 squared is 36. Adding them up: 9 + 4 is 13, plus 36 is 49. Square root of 49 is 7. Yes, that's correct.

Then, ||a × b|| = 7. Then, using the formula:

7 = ||a|| ||b|| sinθ => 7 = 2 * 7 * sinθ => sinθ = 7 / 14 = 1/2. Correct.

So θ = arcsin(1/2). The principal value of arcsin(1/2) is 30°, but since sine is positive in both first and second quadrants, the angle could be 30° or 150°. However, the angle between two vectors is always taken as the smallest angle between them, which is between 0° and 180°, so the smallest possible angle is 30°, and 150° is the obtuse angle. Therefore, the answer should be 30 degrees.

But wait, the problem says "the smallest possible angle between a and b". Is there a scenario where the angle could be smaller? For example, if there are multiple possible angles? But given that sinθ is fixed, θ is uniquely determined up to the angle's supplement. However, since angles between vectors are defined as the smallest one between them, so θ is between 0° and 180°, so the smallest possible angle is 30°, and the other possibility is 150°, which is larger. Therefore, 30° is the answer.

But let me think again. Is there a possibility that the cross product's magnitude could be different? For example, if the cross product vector is given, does that affect anything else? For example, could there be different vectors a and b with the same magnitudes and cross product vector but different angles? Wait, no. Because the magnitude of the cross product is fixed by the formula ||a × b|| = ||a|| ||b|| sinθ. So if the cross product's magnitude is fixed, then sinθ is fixed. Therefore, θ is fixed up to the angle's supplement. But since the angle between vectors is defined as the smallest one, it's 30°.

Alternatively, maybe there's a mistake in assuming that the cross product's magnitude is 7? Let me check again. The cross product vector is (3, 2, 6). So, the magnitude is sqrt(3² + 2² + 6²) = sqrt(9 + 4 + 36) = sqrt(49) = 7. Yes, that's correct.

So, then sinθ = 1/2. Therefore, θ is 30° or 150°, and the smallest angle is 30°. Therefore, the answer is 30 degrees.

But let me check if there's another way to approach this problem. For example, using the dot product. The dot product formula is:

a · b = ||a|| ||b|| cosθ

But we don't have the dot product given. However, if we can find the dot product, we could find cosθ. But since we don't have the dot product, maybe we can relate it with the cross product?

Alternatively, we know that for any two vectors, the magnitude of the cross product squared plus the square of the dot product equals the product of the magnitudes squared times (sin²θ + cos²θ) = ||a||² ||b||².

So, ||a × b||² + (a · b)² = ||a||² ||b||².

Let me check that.

Yes, because:

||a × b||² + (a · b)² = (||a|| ||b|| sinθ)^2 + (||a|| ||b|| cosθ)^2 = ||a||² ||b||² (sin²θ + cos²θ) = ||a||² ||b||².

So, that's an identity.

So, if I compute ||a × b||², which is 7² = 49, and ||a||² ||b||² is 2² * 7² = 4 * 49 = 196. Therefore, (a · b)^2 = 196 - 49 = 147. Therefore, a · b = ±sqrt(147).

But sqrt(147) is sqrt(49*3) = 7*sqrt(3). Therefore, a · b = ±7√3.

Then, since a · b = ||a|| ||b|| cosθ, so:

±7√3 = 2 * 7 * cosθ => ±√3 = 2 cosθ => cosθ = ±√3 / 2.

So, cosθ is either √3 / 2 or -√3 / 2.

Therefore, θ is either 30° or 150°, since cos(30°) = √3 / 2 and cos(150°) = -√3 / 2.

Therefore, again, the angle between the vectors is either 30° or 150°, so the smallest possible angle is 30°.

Therefore, this confirms the previous result.

So, both methods lead to the same conclusion. Therefore, the answer is 30 degrees.

But let me think again if there's any other consideration. For example, could there be a case where the cross product vector is given, but the angle is different? For example, if the cross product vector is fixed, does that impose any constraints on the angle?

Wait, the cross product vector's direction is perpendicular to both a and b. So, the direction of a × b is determined by the right-hand rule. However, the problem doesn't specify any particular direction, just the vector. But since we are only given the magnitude of the cross product, which we already used, and the direction is not affecting the magnitude. Therefore, regardless of the direction of the cross product, the magnitude is fixed, so sinθ is fixed. Therefore, θ is fixed as 30° or 150°, and the smallest angle is 30°.

Therefore, I think the answer is 30 degrees.

But let me check if there's a possible mistake in assuming that the angle between vectors is always between 0° and 180°. Yes, by definition, the angle between two vectors is the smallest angle between them when they are placed tail to tail, so it's always between 0° and 180°, so 30° is indeed the smallest possible angle.

Alternatively, if someone thought that the angle could be negative or something, but no, angles between vectors are always non-negative and less than or equal to 180°.

Therefore, I think I can be confident that the answer is 30 degrees.

**Final Answer**
The smallest possible angle between vectors $\mathbf{a}$ and $\mathbf{b}$ is $\boxed{30}$ degrees.

To determine the smallest possible angle between two vectors $\mathbf{a}$ and $\mathbf{b}$, we are given the following:

- $\|\mathbf{a}\| = 2$
- $\|\mathbf{b}\| = 7$
- $\mathbf{a} \times \mathbf{b} = \begin{pmatrix} 3 \\ 2 \\ 6 \end{pmatrix}$

---

### Step 1: Use the Cross Product Magnitude Formula

The magnitude of the cross product of two vectors is given by:

$$
\|\mathbf{a} \times \mathbf{b}\| = \|\mathbf{a}\| \|\mathbf{b}\| \sin\theta
$$

We are given the vector $\mathbf{a} \times \mathbf{b} = \begin{pmatrix} 3 \\ 2 \\ 6 \end{pmatrix}$, so we can compute its magnitude:

$$
\|\mathbf{a} \times \mathbf{b}\| = \sqrt{3^2 + 2^2 + 6^2} = \sqrt{9 + 4 + 36} = \sqrt{49} = 7
$$

Now plug into the formula:

$$
7 = 2 \cdot 7 \cdot \sin\theta \Rightarrow \sin\theta = \frac{7}{14} = \frac{1}{2}
$$

---

### Step 2: Determine the Angle $\theta$

We now solve for $\theta$ such that $\sin\theta = \frac{1}{2}$. The solutions in the interval $[0^\circ, 180^\circ]$ are:

$$
\theta = 30^\circ \quad \text{or} \quad \theta = 150^\circ
$$

Since the angle between two vectors is defined as the **smallest** angle between them, the smallest possible angle is:

$$
\boxed{30}
$$

---

### Step 3: Confirm with the Dot Product (Optional Verification)

We can also use the identity:

$$
\|\mathbf{a} \times \mathbf{b}\|^2 + (\mathbf{a} \cdot \mathbf{b})^2 = \|\mathbf{a}\|^2 \|\mathbf{b}\|^2
$$

We already know:

- $\|\mathbf{a} \times \mathbf{b}\| = 7 \Rightarrow \|\mathbf{a} \times \mathbf{b}\|^2 = 49$
- $\|\mathbf{a}\|^2 = 4$, $\|\mathbf{b}\|^2 = 49 \Rightarrow \|\mathbf{a}\|^2 \|\mathbf{b}\|^2 = 196$

So:

$$
(\mathbf{a} \cdot \mathbf{b})^2 = 196 - 49 = 147 \Rightarrow \mathbf{a} \cdot \mathbf{b} = \pm \sqrt{147} = \pm 7\sqrt{3}
$$

Now use the dot product formula:

$$
\mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \|\mathbf{b}\| \cos\theta = 2 \cdot 7 \cdot \cos\theta = 14 \cos\theta
$$

So:

$$
14 \cos\theta = \pm 7\sqrt{3} \Rightarrow \cos\theta = \pm \frac{\sqrt{3}}{2}
$$

This gives:

- $\cos\theta = \frac{\sqrt{3}}{2} \Rightarrow \theta = 30^\circ$
- $\cos\theta = -\frac{\sqrt{3}}{2} \Rightarrow \theta = 150^\circ$

Again, the smallest angle is:

$$
\boxed{30}
$$

---

### Final Answer

$$
\boxed{30}
$$

Solution:
Let $\theta$ be the angle between $\mathbf{a}$ and $\mathbf{b}.$  Then
\[\|\mathbf{a} \times \mathbf{b}\| = \|\mathbf{a}\| \|\mathbf{b}\| \sin \theta.\]From the given information, $7 = 14 \sin \theta,$ so $\sin \theta = \frac{1}{2}.$  Thus, the smallest possible value of $\theta$ is $\boxed{30^\circ}.$


Explanation:
Okay, so I need to find the length of the third side of a triangle when two sides are 1 unit and 3 units, and the third side is an integer. Hmm, let me think. I remember there's something called the triangle inequality theorem that might help here. Let me recall what that is.

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. So, for any triangle with sides a, b, and c, the following must all be true:

a + b > c
a + c > b
b + c > a

Right? So, applying this to the given problem where two sides are 1 and 3, and the third side is an integer, let me denote the third side as x. So, the sides are 1, 3, and x.

Let me write down the inequalities:

First inequality: 1 + 3 > x
Second inequality: 1 + x > 3
Third inequality: 3 + x > 1

Let me simplify each of these.

Starting with the first one: 1 + 3 > x
That simplifies to 4 > x, or x < 4.

Second inequality: 1 + x > 3
Subtracting 1 from both sides gives x > 2.

Third inequality: 3 + x > 1
Subtracting 3 from both sides gives x > -2.

But since we're talking about lengths, x has to be a positive number. So, the third inequality doesn't really add any new information because x is already positive. So, combining the first two inequalities, we have x must be greater than 2 and less than 4.

But the problem says that the third side is an integer. So, x has to be an integer. Let me think about the integers between 2 and 4.

Well, integers greater than 2 and less than 4. The integers between 2 and 4 are 3. Wait, because 2 < x < 4. So, x can be 3? But wait, 3 is an integer. But let me check if that's valid.

Wait, but if x is 3, then the sides are 1, 3, 3. Let me check the triangle inequalities again.

First: 1 + 3 > 3? 4 > 3, which is true.
Second: 1 + 3 > 3? Same as above.
Third: 3 + 3 > 1? 6 > 1, which is true.
So, 3 is valid.

But wait, what about x = 2? Let me check. If x is 2, then sides are 1, 2, 3. Let me check the triangle inequalities.

First: 1 + 2 > 3? 3 > 3? No, 3 is not greater than 3. It's equal. So, that's not valid. So, x can't be 2.

Similarly, if x is 4, then sides are 1, 3, 4. Let's check.

First: 1 + 3 > 4? 4 > 4? No, again equal. So, that's not valid.

So, the only integer between 2 and 4 is 3. Therefore, the third side must be 3 units.

Wait, but let me make sure I didn't miss any other integers. Since x has to be greater than 2 and less than 4, the integers in that interval are only 3. So, yes, that's the only possibility.

But let me think again. Maybe I made a mistake in the inequalities? Let me check again.

Original inequalities:

1 + 3 > x => x < 4
1 + x > 3 => x > 2
3 + x > 1 => x > -2 (which is always true since x is positive)

So, combining x > 2 and x < 4, so x must be in (2, 4). Since x is an integer, the only integer in that interval is 3.

Therefore, the third side must be 3 units.

But wait, let me think if there's another way to approach this. Maybe using the concept of triangle sides? For example, the third side must be less than the sum of the other two sides and more than the difference of the other two sides.

Wait, that's another version of the triangle inequality. The third side must be greater than the difference of the other two sides and less than the sum.

So, if the two sides are 1 and 3, then the third side must be greater than 3 - 1 = 2 and less than 3 + 1 = 4. Which is the same as before. So, again, x must be between 2 and 4. So, integer value is 3.

Therefore, the answer is 3.

But let me check once more. Suppose someone might think that maybe 1, 3, and 2 can form a triangle? But as we saw earlier, 1 + 2 is equal to 3, which doesn't satisfy the triangle inequality. So, it's a degenerate triangle, which is essentially a straight line, not a valid triangle. So, 2 is invalid. Similarly, 4 would make 1 + 3 = 4, which is also degenerate.

Therefore, only 3 is valid.

Wait, but let me think if there's a possibility of another integer? For example, if the third side is 1? Then sides would be 1, 1, 3. Let me check triangle inequality. 1 + 1 > 3? 2 > 3? No. So, invalid. Similarly, if third side is 0? Not possible, since length can't be zero.

If third side is 5? Then 1 + 3 = 4 < 5, which violates the triangle inequality. So, no.

So, yeah, only 3 is possible.

Therefore, the length of the third side is 3 units.

**Final Answer**
The length of the third side is \boxed{3} units.

To determine the length of the third side of a triangle with two known sides measuring 1 unit and 3 units, and the third side being an integer, we apply the **Triangle Inequality Theorem**. This theorem states that the sum of the lengths of any two sides of a triangle must be **greater than** the length of the third side.

---

### Step 1: Define the unknown side

Let the length of the third side be $ x $, where $ x $ is an integer. The triangle has sides of lengths:

- $ a = 1 $
- $ b = 3 $
- $ c = x $

---

### Step 2: Apply the Triangle Inequality Theorem

We apply the three inequalities:

1. $ a + b > c $ → $ 1 + 3 > x $ → $ 4 > x $ → $ x < 4 $
2. $ a + c > b $ → $ 1 + x > 3 $ → $ x > 2 $
3. $ b + c > a $ → $ 3 + x > 1 $ → $ x > -2 $ (always true for positive lengths)

Combining the valid inequalities:

$$
2 < x < 4
$$

---

### Step 3: Find integer values in the range

The only integer that satisfies $ 2 < x < 4 $ is:

$$
x = 3
$$

---

### Step 4: Verify the solution

Let’s check if a triangle with sides 1, 3, and 3 is valid:

- $ 1 + 3 > 3 $ → $ 4 > 3 $ ✅
- $ 1 + 3 > 3 $ → $ 4 > 3 $ ✅
- $ 3 + 3 > 1 $ → $ 6 > 1 $ ✅

All triangle inequalities are satisfied, confirming that the triangle is valid.

---

### Final Answer

$$
\boxed{3}
$$

[Assistant]
Explanation:
Okay, so I need to find the length of the third side of a triangle when two sides are 1 unit and 3 units, and the third side is an integer. Hmm, let me think. I remember there's something called the triangle inequality theorem that might help here. Let me recall what that is.

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. So, for any triangle with sides a, b, and c, the following must all be true:

a + b > c
a + c > b
b + c > a

Right? So, applying this to the given problem where two sides are 1 and 3, and the third side is an integer, let me denote the third side as x. So, the sides are 1, 3, and x.

Let me write down the inequalities:

First inequality: 1 + 3 > x
Second inequality: 1 + x > 3
Third inequality: 3 + x > 1

Let me simplify each of these.

Starting with the first one: 1 + 3 > x
That simplifies to 4 > x, or x < 4.

Second inequality: 1 + x > 3
Subtracting 1 from both sides gives x > 2.

Third inequality: 3 + x > 1
Subtracting 3 from both sides gives x > -2.

But since we're talking about lengths, x has to be a positive number. So, the third inequality doesn't really add any new information because x is already positive. So, combining the first two inequalities, we have x must be greater than 2 and less than 4.

But the problem says that the third side is an integer. So, x has to be an integer. Let me think about the integers between 2 and 4.

Well, integers greater than 2 and less than 4. The integers between 2 and 4 are 3. Wait, because 2 < x < 4. So, x can be 3? But wait, 3 is an integer. But let me check if that's valid.

Wait, but if x is 3, then the sides are 1, 3, 3. Let me check the triangle inequalities again.

First: 1 + 3 > 3? 4 > 3, which is true.
Second: 1 + 3 > 3? Same as above.
Third: 3 + 3 > 1? 6 > 1, which is true.
So, 3 is valid.

But wait, what about x = 2? Let me check. If x is 2, then sides are 1, 2, 3. Let me check the triangle inequalities.

First: 1 + 2 > 3? 3 > 3? No, 3 is not greater than 3. It's equal. So, that's not valid. So, x can't be 2.

Similarly, if x is 4, then sides are 1, 3, 4. Let's check.

First: 1 + 3 > 4? 4 > 4? No, again equal. So, that's not valid.

So, the only integer between 2 and 4 is 3. Therefore, the third side must be 3 units.

Wait, but let me make sure I didn't miss any other integers. Since x has to be greater than 2 and less than 4, the integers in that interval are only 3. So, yes, that's the only possibility.

But let me think again. Maybe I made a mistake in the inequalities? Let me check again.

Original inequalities:

1 + 3 > x => x < 4
1 + x > 3 => x > 2
3 + x > 1 => x > -2 (which is always true since x is positive)

So, combining x > 2 and x < 4, so x must be in (2, 4). Since x is an integer, the only integer in that interval is 3.

Therefore, the third side must be 3 units.

But wait, let me think if there's another way to approach this. Maybe using the concept of triangle sides? For example, the third side must be less than the sum of the other two sides and more than the difference of the other two sides.

Wait, that's another version of the triangle inequality. The third side must be greater than the difference of the other two sides and less than the sum.

So, if the two sides are 1 and 3, then the third side must be greater than 3 - 1 = 2 and less than 3 + 1 = 4. Which is the same as before. So, again, x must be between 2 and 4. So, integer value is 3.

Therefore, the answer is 3.

But let me check once more. Suppose someone might think that maybe 1, 3, and 2 can form a triangle? But as we saw earlier, 1 + 2 is equal to 3, which doesn't satisfy the triangle inequality. So, it's a degenerate triangle, which is essentially a straight line, not a valid triangle. So, 2 is invalid. Similarly, 4 would make 1 + 3 = 4, which is also degenerate.

Therefore, only 3 is valid.

Wait, but let me think if there's a possibility of another integer? For example, if the third side is 1? Then sides would be 1, 1, 3. Let me check triangle inequality. 1 + 1 > 3? 2 > 3? No. So, invalid. Similarly, if third side is 0? Not possible, since length can't be zero.

If third side is 5? Then 1 + 3 = 4 < 5, which violates the triangle inequality. So, no.

So, yeah, only 3 is possible.

Therefore, the length of the third side is 3 units.

**Final Answer**
The length of the third side is \boxed{3} units.

To determine the length of the third side of a triangle with two known sides measuring 1 unit and 3 units, and the third side being an integer, we apply the **Triangle Inequality Theorem**. This theorem states that the sum of the lengths of any two sides of a triangle must be **greater than** the length of the third side.

---

### Step 1: Define the unknown side

Let the length of the third side be $ x $, where $ x $ is an integer. The triangle has sides of lengths:

- $ a = 1 $
- $ b = 3 $
- $ c = x $

---

### Step 2: Apply the Triangle Inequality Theorem

We apply the three inequalities:

1. $ a + b > c $ → $ 1 + 3 > x $ → $ 4 > x $ → $ x < 4 $
2. $ a + c > b $ → $ 1 + x > 3 $ → $ x > 2 $
3. $ b + c > a $ → $ 3 + x > 1 $ → $ x > -2 $ (always true for positive lengths)

Combining the valid inequalities:

$$
2 < x < 4
$$

---

### Step 3: Find integer values in the range

The only integer that satisfies $ 2 < x < 4 $ is:

$$
x = 3
$$

---

### Step 4: Verify the solution

Let’s check if a triangle with sides 1, 3, and 3 is valid:

- $ 1 + 3 > 3 $ → $ 4 > 3 $ ✅
- $ 1 + 3 > 3 $ → $ 4 > 3 $ ✅
- $ 3 + 3 > 1 $ → $ 6 > 1 $ ✅

All triangle inequalities are satisfied, confirming that the triangle is valid.

---

### Final Answer

$$
\boxed{3}
$$

Solution: Call the length of the third side $n$. By the triangle inequality, $1+3>n$ and $1+n>3$, or $2<n<4$. The only integer $n$ which satisfies this is $\boxed{3}$.


Explanation:
Okay, let me try to figure out this problem. So, there are twelve friends who went to dinner, and each ordered one meal. But the portions were so large that there was enough food for 18 people. Now, the question is, if they share, how many meals should they have ordered to have just enough food for the 12 of them?

Hmm, let me break this down. First, they ordered 12 meals, but the food was enough for 18 people. That means each meal they ordered actually provided enough food for more than one person. Wait, how does that work? Maybe each meal is portioned for a certain number of people? Let me think.

If 12 meals were enough for 18 people, then each meal must be providing food for 18 divided by 12 people. Let me write that down. So, the amount of food per meal is 18 people / 12 meals. Let me calculate that. 18 divided by 12 is 1.5. So each meal is equivalent to 1.5 people's worth of food? That seems like a lot, but maybe the portions are really large.

So, if each meal is enough for 1.5 people, then if they want to have just enough food for 12 people, how many meals do they need? Let me think. If one meal is 1.5 people, then the number of meals needed for 12 people would be 12 divided by 1.5. Let me do that calculation. 12 divided by 1.5. Hmm, 1.5 times 8 is 12, right? Because 1.5 times 2 is 3, times 4 is 6, times 8 is 12. So that would be 8 meals. Wait, so they should have ordered 8 meals instead of 12?

But let me check again. If 12 meals are enough for 18 people, then the total amount of food is equivalent to 18 people. So if they want to have enough food for 12 people, they need to order meals such that the total food equals 12 people. Since 12 meals give 18 people, then the ratio of meals to people is 12 meals : 18 people. So, simplifying that ratio, divide both by 6: 2 meals : 3 people. So, for every 2 meals, you get food for 3 people. Therefore, to get food for 12 people, how many meals?

Set up a proportion. Let me write that as:

2 meals / 3 people = x meals / 12 people

Cross-multiplying, 3x = 24, so x = 8. Yep, same answer. So they should have ordered 8 meals. Therefore, the answer is 8.

But let me think again to make sure I didn't make a mistake. If each meal is for 1.5 people, then 8 meals would be 8 * 1.5 = 12 people. Exactly. So that works. So if they ordered 8 meals, the food would be enough for 12 people. But originally, they ordered 12 meals, which was enough for 18 people. So, the answer is 8 meals.

Alternatively, maybe think in terms of how much food they have. The total food they ordered was enough for 18 people. But they only needed food for 12 people. So, they have extra food. But the question is, how many meals should they have ordered to have just enough food for 12 people? So, if the total food required for 12 people is less than what 12 meals provide, so they need fewer meals.

So, the total food is equivalent to 18 people. So, if they want to have food for 12 people, they need to order meals such that the total food is 12 people. So, how many meals would that be? Since 12 meals = 18 people, then 1 meal = 18/12 = 1.5 people. Therefore, 12 people would need 12 / 1.5 = 8 meals. Yep, same answer again.

Another way: think of it as a rate. The amount of food per meal. If 12 meals provide food for 18 people, then the amount of food per meal is 18 people / 12 meals = 1.5 people per meal. Therefore, to get food for 12 people, divide 12 by 1.5, which is 8.

Alternatively, think of it as scaling. If they want to reduce the number of people from 18 to 12, that's a scaling factor of 12/18 = 2/3. Therefore, the number of meals should also be scaled by 2/3. So, original meals were 12, so 12 * (2/3) = 8.

All these different approaches lead me to the same answer, 8. So I think that's correct.

But let me check if there's another way someone might approach it and get a different answer, just to be thorough. For example, maybe someone might think that since they had 12 meals and there was enough for 18 people, then each person ate 1.5 meals? Wait, that might be a different interpretation. Wait, if there are 12 people, and the food was enough for 18 people, then each person had enough food for 1.5 people? Wait, that might be confusing.

Wait, maybe the confusion is about whether the food was enough for 18 people in total, or if each person had enough for 1.5 people. But the problem says "there was enough food for 18 people." So, the total food was sufficient for 18 people. So, if there are 12 people, they would have more than enough. But if they want to have just enough for 12 people, how many meals should they order?

So, if the total food for 12 meals is enough for 18 people, then the total food is equivalent to 18 people. Therefore, to have food for 12 people, they need to order meals such that the total food is 12 people. So, since 12 meals = 18 people, then 1 meal = 1.5 people. Therefore, 12 people would need 12 / 1.5 = 8 meals.

Alternatively, if you think of it as the amount of food per person. If 12 meals are enough for 18 people, then each person gets 12 meals / 18 people = 2/3 of a meal per person. Therefore, for 12 people, they need 12 people * (2/3 meal/person) = 8 meals.

Yes, that's another way to think about it. So, each person gets 2/3 of a meal. Therefore, 12 people would need 8 meals.

So, all these different methods confirm that the answer is 8. Therefore, I think that's correct.

I don't see any other way to interpret the problem that would lead to a different answer. The key is understanding that the total amount of food is proportional to the number of meals, and that the original 12 meals provided enough for 18 people, so the ratio of meals to people is 12:18, which simplifies to 2:3. Therefore, to get the number of meals needed for 12 people, set up the proportion 2 meals / 3 people = x meals / 12 people, solve for x, which is 8.

Therefore, the answer is 8 meals.

**Final Answer**
The twelve friends should have ordered \boxed{8} meals.

To determine how many meals the twelve friends should have ordered to have just enough food for all of them, we begin by analyzing the relationship between the number of meals and the amount of food they provided.

---

### Step 1: Understand the Given Information

- **12 meals** were ordered.
- These meals were **enough for 18 people**.
- This implies that the **total food** from 12 meals is equivalent to the food needed for 18 people.

---

### Step 2: Determine the Food per Meal

We can calculate how much food each meal provides by dividing the total number of people the food can feed by the number of meals:

$$
\text{Food per meal} = \frac{18 \text{ people}}{12 \text{ meals}} = 1.5 \text{ people per meal}
$$

So, **each meal provides enough food for 1.5 people**.

---

### Step 3: Calculate the Number of Meals Needed for 12 People

Now, we want to find out how many meals are needed to provide food for **12 people**, given that each meal provides food for 1.5 people:

$$
\text{Number of meals} = \frac{12 \text{ people}}{1.5 \text{ people per meal}} = 8 \text{ meals}
$$

---

### Step 4: Verify the Result

To double-check, we can use a proportion:

$$
\frac{12 \text{ meals}}{18 \text{ people}} = \frac{x \text{ meals}}{12 \text{ people}}
$$

Cross-multiplying:

$$
12x = 18 \times 12 \Rightarrow x = \frac{18 \times 12}{12} = 18 \div 1.5 = 8
$$

This confirms that **8 meals** are sufficient for 12 people.

---

### Final Answer

$$
\boxed{8}
$$

[Assistant]
Explanation:
Okay, let me try to figure out this problem. So, there are twelve friends who went to dinner, and each ordered one meal. But the portions were so large that there was enough food for 18 people. Now, the question is, if they share, how many meals should they have ordered to have just enough food for the 12 of them?

Hmm, let me break this down. First, they ordered 12 meals, but the food was enough for 18 people. That means each meal they ordered actually provided enough food for more than one person. Wait, how does that work? Maybe each meal is portioned for a certain number of people? Let me think.

If 12 meals were enough for 18 people, then each meal must be providing food for 18 divided by 12 people. Let me write that down. So, the amount of food per meal is 18 people / 12 meals. Let me calculate that. 18 divided by 12 is 1.5. So each meal is equivalent to 1.5 people's worth of food? That seems like a lot, but maybe the portions are really large.

So, if each meal is enough for 1.5 people, then if they want to have just enough food for 12 people, how many meals do they need? Let me think. If one meal is 1.5 people, then the number of meals needed for 12 people would be 12 divided by 1.5. Let me do that calculation. 12 divided by 1.5. Hmm, 1.5 times 8 is 12, right? Because 1.5 times 2 is 3, times 4 is 6, times 8 is 12. So that would be 8 meals. Wait, so they should have ordered 8 meals instead of 12?

But let me check again. If 12 meals are enough for 18 people, then the total amount of food is equivalent to 18 people. So if they want to have enough food for 12 people, they need to order meals such that the total food equals 12 people. Since 12 meals give 18 people, then the ratio of meals to people is 12 meals : 18 people. So, simplifying that ratio, divide both by 6: 2 meals : 3 people. So, for every 2 meals, you get food for 3 people. Therefore, to get food for 12 people, how many meals?

Set up a proportion. Let me write that as:

2 meals / 3 people = x meals / 12 people

Cross-multiplying, 3x = 24, so x = 8. Yep, same answer. So they should have ordered 8 meals. Therefore, the answer is 8.

But let me think again to make sure I didn't make a mistake. If each meal is for 1.5 people, then 8 meals would be 8 * 1.5 = 12 people. Exactly. So that works. So if they ordered 8 meals, the food would be enough for 12 people. But originally, they ordered 12 meals, which was enough for 18 people. So, the answer is 8 meals.

Alternatively, maybe think in terms of how much food they have. The total food they ordered was enough for 18 people. But they only needed food for 12 people. So, they have extra food. But the question is, how many meals should they have ordered to have just enough food for 12 people? So, if the total food required for 12 people is less than what 12 meals provide, so they need fewer meals.

So, the total food is equivalent to 18 people. So, if they want to have food for 12 people, they need to order meals such that the total food is 12 people. So, how many meals would that be? Since 12 meals = 18 people, then 1 meal = 18/12 = 1.5 people. Therefore, 12 people would need 12 / 1.5 = 8 meals. Yep, same answer again.

Another way: think of it as a rate. The amount of food per meal. If 12 meals provide food for 18 people, then the amount of food per meal is 18 people / 12 meals = 1.5 people per meal. Therefore, to get food for 12 people, divide 12 by 1.5, which is 8.

Alternatively, think of it as scaling. If they want to reduce the number of people from 18 to 12, that's a scaling factor of 12/18 = 2/3. Therefore, the number of meals should also be scaled by 2/3. So, original meals were 12, so 12 * (2/3) = 8.

All these different approaches lead me to the same answer, 8. So I think that's correct.

But let me check if there's another way someone might approach it and get a different answer, just to be thorough. For example, maybe someone might think that since they had 12 meals and there was enough for 18 people, then each person ate 1.5 meals? Wait, that might be a different interpretation. Wait, if there are 12 people, and the food was enough for 18 people, then each person had enough food for 1.5 people? Wait, that might be confusing.

Wait, maybe the confusion is about whether the food was enough for 18 people in total, or if each person had enough for 1.5 people. But the problem says "there was enough food for 18 people." So, the total food was sufficient for 18 people. So, if there are 12 people, they would have more than enough. But if they want to have just enough for 12 people, how many meals should they order?

So, if the total food for 12 meals is enough for 18 people, then the total food is equivalent to 18 people. Therefore, to have food for 12 people, they need to order meals such that the total food is 12 people. So, since 12 meals = 18 people, then 1 meal = 1.5 people. Therefore, 12 people would need 12 / 1.5 = 8 meals.

Alternatively, if you think of it as the amount of food per person. If 12 meals are enough for 18 people, then each person gets 12 meals / 18 people = 2/3 of a meal per person. Therefore, for 12 people, they need 12 people * (2/3 meal/person) = 8 meals.

Yes, that's another way to think about it. So, each person gets 2/3 of a meal. Therefore, 12 people would need 8 meals.

So, all these different methods confirm that the answer is 8. Therefore, I think that's correct.

I don't see any other way to interpret the problem that would lead to a different answer. The key is understanding that the total amount of food is proportional to the number of meals, and that the original 12 meals provided enough for 18 people, so the ratio of meals to people is 12:18, which simplifies to 2:3. Therefore, to get the number of meals needed for 12 people, set up the proportion 2 meals / 3 people = x meals / 12 people, solve for x, which is 8.

Therefore, the answer is 8 meals.

**Final Answer**
The twelve friends should have ordered \boxed{8} meals.

To determine how many meals the twelve friends should have ordered to have just enough food for all of them, we begin by analyzing the relationship between the number of meals and the amount of food they provided.

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### Step 1: Understand the Given Information

- **12 meals** were ordered.
- These meals were **enough for 18 people**.
- This implies that the **total food** from 12 meals is equivalent to the food needed for 18 people.

---

### Step 2: Determine the Food per Meal

We can calculate how much food each meal provides by dividing the total number of people the food can feed by the number of meals:

$$
\text{Food per meal} = \frac{18 \text{ people}}{12 \text{ meals}} = 1.5 \text{ people per meal}
$$

So, **each meal provides enough food for 1.5 people**.

---

### Step 3: Calculate the Number of Meals Needed for 12 People

Now, we want to find out how many meals are needed to provide food for **12 people**, given that each meal provides food for 1.5 people:

$$
\text{Number of meals} = \frac{12 \text{ people}}{1.5 \text{ people per meal}} = 8 \text{ meals}
$$

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### Step 4: Verify the Result

To double-check, we can use a proportion:

$$
\frac{12 \text{ meals}}{18 \text{ people}} = \frac{x \text{ meals}}{12 \text{ people}}
$$

Cross-multiplying:

$$
12x = 18 \times 12 \Rightarrow x = \frac{18 \times 12}{12} = 18 \div 1.5 = 8
$$

This confirms that **8 meals** are sufficient for 12 people.

---

### Final Answer

$$
\boxed{8}
$$

Solution: If 12 people order $\frac{18}{12}=1\frac{1}{2}$ times too much food, they should have ordered $\frac{12}{\frac{3}{2}}=\frac{2}{3}\times 12=\boxed{8}$ meals.