Explanation:
Okay, so Mikka wants to order a pizza with two different toppings, and he has 8 different toppings to choose from. The question is asking how many different pizzas he could order. Hmm, let me think about this step by step.

First, I need to figure out if this is a combination or permutation problem. Since the order of the toppings doesn't matter on a pizza—meaning that a pizza with pepperoni and mushrooms is the same as one with mushrooms and pepperoni—it should be a combination problem. Combinations are used when the order doesn't matter, whereas permutations are when the order does. So, combinations it is.

Now, the formula for combinations is C(n, k) = n! / (k! * (n - k)!), where n is the total number of items, and k is the number of items to choose. In this case, n is 8 toppings, and k is 2 toppings. So plugging those numbers into the formula, we get C(8, 2) = 8! / (2! * (8 - 2)!).

Let me compute that. First, 8! is 8 factorial, which is 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1. But wait, since we have (8 - 2)! which is 6!, maybe we can simplify the calculation. Let me write that out:

C(8, 2) = 8! / (2! * 6!) = (8 × 7 × 6!) / (2! × 6!)

Oh, right! The 6! in the numerator and denominator cancels out. So that simplifies to (8 × 7) / 2!

Since 2! is 2 × 1 = 2, so then it becomes (8 × 7) / 2. Let me compute that. 8 multiplied by 7 is 56. Then 56 divided by 2 is 28. So, does that mean there are 28 different pizzas he could order?

Wait, let me double-check. Maybe I can think of it another way. If there are 8 toppings, and he wants to choose 2, then for the first topping, he has 8 choices, and for the second topping, he has 7 choices left. But since order doesn't matter, this method would count each pizza twice. For example, choosing pepperoni first and then mushrooms is the same as choosing mushrooms first and then pepperoni. So, the total number of ordered pairs is 8 × 7 = 56, but since each pizza is counted twice, we divide by 2. So 56 / 2 = 28. Yep, same answer. So that seems to confirm it.

Alternatively, maybe I can list them out for a smaller number to check. Let's say there are 3 toppings: A, B, C. How many combinations of 2? It should be 3: AB, AC, BC. Using the formula, C(3, 2) = 3! / (2! * 1!) = 6 / 2 = 3. Correct. If I use the other method: 3 choices for first, 2 for second, so 6, divide by 2, get 3. Correct again. So the formula works here.

So applying that to 8 toppings, 28 seems right. Let me think if there's any other way this could be interpreted. For example, does the problem say "two different toppings," so does that mean that he can't have the same topping twice? Well, since he's choosing two different toppings, that's already implied. So each pizza has exactly two distinct toppings, no repeats. So the answer is 28.

Wait, but just to make sure, maybe the question is trying to trick me? Like, maybe some toppings are the same? But no, the problem says he has 8 different toppings. So each topping is unique. So choosing any two different ones is just combinations. So yeah, 28 is the answer.

Another way to think about it: the number of combinations is the same as the number of edges in a complete graph with 8 vertices. Each vertex represents a topping, and each edge represents a pair of toppings. The number of edges in a complete graph is C(n, 2), which is exactly what we calculated. So again, 28.

I think that's solid. So the answer is 28.

**Final Answer**
The number of different pizzas Mikka could order is \boxed{28}.

To determine how many different pizzas Mikka could order, we need to analyze the situation carefully.

---

### **Understanding the Problem**

Mikka wants to order a pizza with **two different toppings**, and he has **8 different toppings** to choose from. The key point here is that the **order of toppings does not matter** on a pizza. That is, a pizza with **pepperoni and mushrooms** is the same as one with **mushrooms and pepperoni**.

This is a classic **combinations** problem, where we are selecting **2 items** from a set of **8**, and the order of selection does not matter.

---

### **Using the Combination Formula**

The formula for combinations is:

$$
C(n, k) = \frac{n!}{k!(n - k)!}
$$

Where:
- $ n $ is the total number of items (toppings),
- $ k $ is the number of items to choose (toppings per pizza),
- $ ! $ denotes factorial.

In this case:
- $ n = 8 $
- $ k = 2 $

So we compute:

$$
C(8, 2) = \frac{8!}{2!(8 - 2)!} = \frac{8!}{2! \cdot 6!}
$$

We can simplify this by canceling out the $ 6! $ in the numerator and denominator:

$$
C(8, 2) = \frac{8 \times 7 \times 6!}{2 \times 1 \times 6!} = \frac{8 \times 7}{2} = \frac{56}{2} = 28
$$

---

### **Alternative Interpretation**

Another way to think about this is by considering the number of **ordered pairs** of toppings. If Mikka first chooses one of the 8 toppings, and then another of the remaining 7, that gives $ 8 \times 7 = 56 $ ordered pairs. However, since the order doesn't matter, each pizza is counted **twice** (once for each order of the two toppings). So we divide by 2:

$$
\frac{8 \times 7}{2} = 28
$$

This confirms our earlier result.

---

### **Conclusion**

Mikka can order **28 different pizzas**, each with a unique combination of two different toppings from the 8 available.

$$
\boxed{28}
$$

[Assistant]
Explanation:
Okay, so Mikka wants to order a pizza with two different toppings, and he has 8 different toppings to choose from. The question is asking how many different pizzas he could order. Hmm, let me think about this step by step.

First, I need to figure out if this is a combination or permutation problem. Since the order of the toppings doesn't matter on a pizza—meaning that a pizza with pepperoni and mushrooms is the same as one with mushrooms and pepperoni—it should be a combination problem. Combinations are used when the order doesn't matter, whereas permutations are when the order does. So, combinations it is.

Now, the formula for combinations is C(n, k) = n! / (k! * (n - k)!), where n is the total number of items, and k is the number of items to choose. In this case, n is 8 toppings, and k is 2 toppings. So plugging those numbers into the formula, we get C(8, 2) = 8! / (2! * (8 - 2)!).

Let me compute that. First, 8! is 8 factorial, which is 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1. But wait, since we have (8 - 2)! which is 6!, maybe we can simplify the calculation. Let me write that out:

C(8, 2) = 8! / (2! * 6!) = (8 × 7 × 6!) / (2! × 6!)

Oh, right! The 6! in the numerator and denominator cancels out. So that simplifies to (8 × 7) / 2!

Since 2! is 2 × 1 = 2, so then it becomes (8 × 7) / 2. Let me compute that. 8 multiplied by 7 is 56. Then 56 divided by 2 is 28. So, does that mean there are 28 different pizzas he could order?

Wait, let me double-check. Maybe I can think of it another way. If there are 8 toppings, and he wants to choose 2, then for the first topping, he has 8 choices, and for the second topping, he has 7 choices left. But since order doesn't matter, this method would count each pizza twice. For example, choosing pepperoni first and then mushrooms is the same as choosing mushrooms first and then pepperoni. So, the total number of ordered pairs is 8 × 7 = 56, but since each pizza is counted twice, we divide by 2. So 56 / 2 = 28. Yep, same answer. So that seems to confirm it.

Alternatively, maybe I can list them out for a smaller number to check. Let's say there are 3 toppings: A, B, C. How many combinations of 2? It should be 3: AB, AC, BC. Using the formula, C(3, 2) = 3! / (2! * 1!) = 6 / 2 = 3. Correct. If I use the other method: 3 choices for first, 2 for second, so 6, divide by 2, get 3. Correct again. So the formula works here.

So applying that to 8 toppings, 28 seems right. Let me think if there's any other way this could be interpreted. For example, does the problem say "two different toppings," so does that mean that he can't have the same topping twice? Well, since he's choosing two different toppings, that's already implied. So each pizza has exactly two distinct toppings, no repeats. So the answer is 28.

Wait, but just to make sure, maybe the question is trying to trick me? Like, maybe some toppings are the same? But no, the problem says he has 8 different toppings. So each topping is unique. So choosing any two different ones is just combinations. So yeah, 28 is the answer.

Another way to think about it: the number of combinations is the same as the number of edges in a complete graph with 8 vertices. Each vertex represents a topping, and each edge represents a pair of toppings. The number of edges in a complete graph is C(n, 2), which is exactly what we calculated. So again, 28.

I think that's solid. So the answer is 28.

**Final Answer**
The number of different pizzas Mikka could order is \boxed{28}.

To determine how many different pizzas Mikka could order, we need to analyze the situation carefully.

---

### **Understanding the Problem**

Mikka wants to order a pizza with **two different toppings**, and he has **8 different toppings** to choose from. The key point here is that the **order of toppings does not matter** on a pizza. That is, a pizza with **pepperoni and mushrooms** is the same as one with **mushrooms and pepperoni**.

This is a classic **combinations** problem, where we are selecting **2 items** from a set of **8**, and the order of selection does not matter.

---

### **Using the Combination Formula**

The formula for combinations is:

$$
C(n, k) = \frac{n!}{k!(n - k)!}
$$

Where:
- $ n $ is the total number of items (toppings),
- $ k $ is the number of items to choose (toppings per pizza),
- $ ! $ denotes factorial.

In this case:
- $ n = 8 $
- $ k = 2 $

So we compute:

$$
C(8, 2) = \frac{8!}{2!(8 - 2)!} = \frac{8!}{2! \cdot 6!}
$$

We can simplify this by canceling out the $ 6! $ in the numerator and denominator:

$$
C(8, 2) = \frac{8 \times 7 \times 6!}{2 \times 1 \times 6!} = \frac{8 \times 7}{2} = \frac{56}{2} = 28
$$

---

### **Alternative Interpretation**

Another way to think about this is by considering the number of **ordered pairs** of toppings. If Mikka first chooses one of the 8 toppings, and then another of the remaining 7, that gives $ 8 \times 7 = 56 $ ordered pairs. However, since the order doesn't matter, each pizza is counted **twice** (once for each order of the two toppings). So we divide by 2:

$$
\frac{8 \times 7}{2} = 28
$$

This confirms our earlier result.

---

### **Conclusion**

Mikka can order **28 different pizzas**, each with a unique combination of two different toppings from the 8 available.

$$
\boxed{28}
$$

Solution: There are 8 choices of toppings, and we need to choose 2 distinct toppings. This is represented by the number of 2-element subsets of an 8 -element set. We use the binomial coefficient ${8 \choose 2} = \boxed{28}$ to compute this.


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 what 5^5 divided by 5^4 minus 5^3 plus 5^2 times 5^1 is as an integer. Let me write that down first to make sure I have the expression right: 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 simplifies to 5.

Okay, so the first part simplifies to 5. 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 calculate that. 5^3 is 5*5*5, which is 125. So that term is -125.

So 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 entire expression simplifies to 5.

Wait, but let me make sure I didn't make a mistake in the order of operations. The original expression is 5^5 ÷ 5^4 - 5^3 + 5^2 · 5^1. According to PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), we do exponents first, then multiplication and division from left to right, then addition and subtraction from left to right.

So let me go through it again step by step.

First, compute all exponents:

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

So substituting those values into the expression: 3125 ÷ 625 - 125 + 25 · 5.

Now, perform division and multiplication from left to right.

First, 3125 ÷ 625. Let me calculate that. 625 times 5 is 3125, so that's 5.

Then, 25 · 5 is 125.

So now the expression becomes 5 - 125 + 125.

Which is the same as before. So 5 - 125 is -120, then -120 + 125 is 5.

So that seems consistent.

Alternatively, maybe I can think of it as combining the terms. Let me check if there's another way to approach this.

Original expression: 5^5 / 5^4 - 5^3 + 5^2 * 5^1.

As I did before, 5^5 / 5^4 is 5^(5-4) = 5^1 = 5.

Then, 5^2 * 5^1 is 5^(2+1) = 5^3 = 125.

So the expression becomes 5 - 5^3 + 5^3.

Wait, that's 5 - 5^3 + 5^3. Then, the -5^3 and +5^3 cancel each other, leaving just 5.

Oh, that's a simpler way to see it. So that's why the answer is 5.

But let me make sure that I didn't misinterpret the original expression. Is it 5^5 divided by 5^4 minus 5^3 plus 5^2 times 5^1? Or could there be parentheses that change the order? For example, if it was (5^5 ÷ 5^4 - 5^3) + (5^2 · 5^1), which is how it's written.

But according to order of operations, division and multiplication have the same priority and are done left to right. So the original expression is evaluated as (5^5 ÷ 5^4) - 5^3 + (5^2 · 5^1).

So yes, that's how we did it.

Alternatively, if someone misreads the expression, maybe they might think that the subtraction applies to the entire 5^3 + 5^2*5^1, but that would require parentheses. Since there are no parentheses, we follow the standard order.

So, in conclusion, the expression simplifies to 5.

But just to be thorough, let me compute all the terms numerically without simplifying exponents first.

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

Calculating each term:

5^5 = 5*5*5*5*5. Let me compute that step by step.

5*5 = 25, 25*5 = 125, 125*5 = 625, 625*5 = 3125. So 5^5 is 3125.

5^4 is 5*5*5*5. 5*5=25, 25*5=125, 125*5=625. So 5^4 is 625.

So 3125 ÷ 625 is 5, as before.

Then, 5^3 is 125.

5^2 is 25, 5^1 is 5.

So 25*5 is 125.

So substituting back: 5 - 125 + 125.

So 5 - 125 is -120, then -120 + 125 is 5.

Yep, same result.

Alternatively, if I do all the operations step by step:

Start with 5^5 ÷ 5^4. That's 3125 ÷ 625 = 5.

Then subtract 5^3: 5 - 125 = -120.

Then add 5^2 · 5^1: -120 + 125 = 5.

So that's consistent.

Another way to think about it: maybe factor out some terms?

Looking at the original expression: 5^5 / 5^4 - 5^3 + 5^2 * 5^1.

We can write this as 5^(5-4) - 5^3 + 5^(2+1) = 5^1 - 5^3 + 5^3.

Which is 5 - 5^3 + 5^3.

Then, the -5^3 and +5^3 cancel each other, leaving 5.

That's a much more straightforward way.

So regardless of the method, it seems like the answer is 5.

But just to make sure, let me check if there's any chance I misread the original problem.

The problem says: 5^5 ÷ 5^4 - 5^3 + 5^2 · 5^1.

Yes, that's exactly what I worked with.

Another possible mistake could be if someone does the operations in the wrong order. For example, if someone does 5^5 ÷ (5^4 - 5^3) + 5^2 · 5^1, but that would require parentheses, which aren't there.

Alternatively, if someone does 5^5 ÷ 5^4 - (5^3 + 5^2) · 5^1, but again, that would need parentheses.

But since there are no parentheses, we follow the standard order.

So, I think I can be confident that the answer is 5.

Just to recap:

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

Step 1: Simplify exponents and division/multiplication left to right.

5^5 ÷ 5^4 = 5^(5-4) = 5^1 = 5

5^2 · 5^1 = 5^(2+1) = 5^3 = 125

Now the expression is 5 - 5^3 + 125

Compute 5^3 = 125, so:

5 - 125 + 125 = 5 + (-125 + 125) = 5 + 0 = 5

Yep, that's correct.

Alternatively, if I think of it as:

5 - 125 + 125 = 5 + ( -125 + 125 ) = 5 + 0 = 5

So, all steps check out.

Therefore, the final answer 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: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

---

### Step 1: Simplify the division and multiplication

We start by simplifying the terms involving exponents using the rules of exponents:

- **Division of exponents with the same base:**
  $$
  5^5 \div 5^4 = 5^{5-4} = 5^1 = 5
  $$

- **Multiplication of exponents with the same base:**
  $$
  5^2 \cdot 5^1 = 5^{2+1} = 5^3 = 125
  $$

So the expression now becomes:

$$
5 - 5^3 + 125
$$

---

### Step 2: Evaluate the remaining exponent

$$
5^3 = 125
$$

Substitute this back into the expression:

$$
5 - 125 + 125
$$

---

### Step 3: Perform the arithmetic

Now, simplify from left to right:

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

---

### Final Answer

$$
\boxed{5}
$$

[Assistant]
Explanation:
Okay, so I need to figure out what 5^5 divided by 5^4 minus 5^3 plus 5^2 times 5^1 is as an integer. Let me write that down first to make sure I have the expression right: 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 simplifies to 5.

Okay, so the first part simplifies to 5. 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 calculate that. 5^3 is 5*5*5, which is 125. So that term is -125.

So 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 entire expression simplifies to 5.

Wait, but let me make sure I didn't make a mistake in the order of operations. The original expression is 5^5 ÷ 5^4 - 5^3 + 5^2 · 5^1. According to PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), we do exponents first, then multiplication and division from left to right, then addition and subtraction from left to right.

So let me go through it again step by step.

First, compute all exponents:

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

So substituting those values into the expression: 3125 ÷ 625 - 125 + 25 · 5.

Now, perform division and multiplication from left to right.

First, 3125 ÷ 625. Let me calculate that. 625 times 5 is 3125, so that's 5.

Then, 25 · 5 is 125.

So now the expression becomes 5 - 125 + 125.

Which is the same as before. So 5 - 125 is -120, then -120 + 125 is 5.

So that seems consistent.

Alternatively, maybe I can think of it as combining the terms. Let me check if there's another way to approach this.

Original expression: 5^5 / 5^4 - 5^3 + 5^2 * 5^1.

As I did before, 5^5 / 5^4 is 5^(5-4) = 5^1 = 5.

Then, 5^2 * 5^1 is 5^(2+1) = 5^3 = 125.

So the expression becomes 5 - 5^3 + 5^3.

Wait, that's 5 - 5^3 + 5^3. Then, the -5^3 and +5^3 cancel each other, leaving just 5.

Oh, that's a simpler way to see it. So that's why the answer is 5.

But let me make sure that I didn't misinterpret the original expression. Is it 5^5 divided by 5^4 minus 5^3 plus 5^2 times 5^1? Or could there be parentheses that change the order? For example, if it was (5^5 ÷ 5^4 - 5^3) + (5^2 · 5^1), which is how it's written.

But according to order of operations, division and multiplication have the same priority and are done left to right. So the original expression is evaluated as (5^5 ÷ 5^4) - 5^3 + (5^2 · 5^1).

So yes, that's how we did it.

Alternatively, if someone misreads the expression, maybe they might think that the subtraction applies to the entire 5^3 + 5^2*5^1, but that would require parentheses. Since there are no parentheses, we follow the standard order.

So, in conclusion, the expression simplifies to 5.

But just to be thorough, let me compute all the terms numerically without simplifying exponents first.

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

Calculating each term:

5^5 = 5*5*5*5*5. Let me compute that step by step.

5*5 = 25, 25*5 = 125, 125*5 = 625, 625*5 = 3125. So 5^5 is 3125.

5^4 is 5*5*5*5. 5*5=25, 25*5=125, 125*5=625. So 5^4 is 625.

So 3125 ÷ 625 is 5, as before.

Then, 5^3 is 125.

5^2 is 25, 5^1 is 5.

So 25*5 is 125.

So substituting back: 5 - 125 + 125.

So 5 - 125 is -120, then -120 + 125 is 5.

Yep, same result.

Alternatively, if I do all the operations step by step:

Start with 5^5 ÷ 5^4. That's 3125 ÷ 625 = 5.

Then subtract 5^3: 5 - 125 = -120.

Then add 5^2 · 5^1: -120 + 125 = 5.

So that's consistent.

Another way to think about it: maybe factor out some terms?

Looking at the original expression: 5^5 / 5^4 - 5^3 + 5^2 * 5^1.

We can write this as 5^(5-4) - 5^3 + 5^(2+1) = 5^1 - 5^3 + 5^3.

Which is 5 - 5^3 + 5^3.

Then, the -5^3 and +5^3 cancel each other, leaving 5.

That's a much more straightforward way.

So regardless of the method, it seems like the answer is 5.

But just to make sure, let me check if there's any chance I misread the original problem.

The problem says: 5^5 ÷ 5^4 - 5^3 + 5^2 · 5^1.

Yes, that's exactly what I worked with.

Another possible mistake could be if someone does the operations in the wrong order. For example, if someone does 5^5 ÷ (5^4 - 5^3) + 5^2 · 5^1, but that would require parentheses, which aren't there.

Alternatively, if someone does 5^5 ÷ 5^4 - (5^3 + 5^2) · 5^1, but again, that would need parentheses.

But since there are no parentheses, we follow the standard order.

So, I think I can be confident that the answer is 5.

Just to recap:

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

Step 1: Simplify exponents and division/multiplication left to right.

5^5 ÷ 5^4 = 5^(5-4) = 5^1 = 5

5^2 · 5^1 = 5^(2+1) = 5^3 = 125

Now the expression is 5 - 5^3 + 125

Compute 5^3 = 125, so:

5 - 125 + 125 = 5 + (-125 + 125) = 5 + 0 = 5

Yep, that's correct.

Alternatively, if I think of it as:

5 - 125 + 125 = 5 + ( -125 + 125 ) = 5 + 0 = 5

So, all steps check out.

Therefore, the final answer 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: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

---

### Step 1: Simplify the division and multiplication

We start by simplifying the terms involving exponents using the rules of exponents:

- **Division of exponents with the same base:**
  $$
  5^5 \div 5^4 = 5^{5-4} = 5^1 = 5
  $$

- **Multiplication of exponents with the same base:**
  $$
  5^2 \cdot 5^1 = 5^{2+1} = 5^3 = 125
  $$

So the expression now becomes:

$$
5 - 5^3 + 125
$$

---

### Step 2: Evaluate the remaining exponent

$$
5^3 = 125
$$

Substitute this back into the expression:

$$
5 - 125 + 125
$$

---

### Step 3: Perform the arithmetic

Now, simplify 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.

So, according to the formula:

||a × b|| = ||a|| ||b|| sinθ

Plugging in the known values:

7 = 2 * 7 * sinθ

Wait, let me check that again. ||a|| is 2, ||b|| is 7, so 2*7 is 14. Therefore:

7 = 14 sinθ

So, solving for sinθ:

sinθ = 7 / 14 = 1/2

Therefore, θ = arcsin(1/2). The arcsin of 1/2 is 30 degrees or 150 degrees. But since we are looking for the smallest possible angle between them, it should be 30 degrees. Wait, but hold on—is there a possibility that the angle could be something else?

Wait, the cross product's magnitude depends on sinθ, and since sinθ is positive in the first and second quadrants, θ could be between 0 and 180 degrees. But the smallest angle between two vectors is always considered to be the acute or right angle, so if θ is obtuse, the smallest angle would be 180 - θ. However, in this case, since sinθ is 1/2, θ could be 30 degrees or 150 degrees. But since the question asks for the smallest possible angle, it's 30 degrees.

But wait, let me make sure that there isn't any other consideration here. The cross product vector's direction is perpendicular to both a and b, but does that affect the angle? Hmm, no, because the magnitude of the cross product only depends on the sine of the angle between them, regardless of their direction. So, regardless of the direction of the cross product, the magnitude is determined by the sine of the angle. Therefore, the calculation seems straightforward.

But let me verify again. Let me recheck the cross product magnitude. The cross product vector is (3, 2, 6). So, the magnitude squared is 3² + 2² + 6² = 9 + 4 + 36 = 49. So, magnitude is 7. Correct. Then, ||a × b|| = 7. Then, 7 = ||a|| ||b|| sinθ => 7 = 2*7*sinθ => sinθ = 1/2. So θ is 30° or 150°, so the smallest angle is 30°. Therefore, the answer is 30 degrees.

But wait, the problem says "the smallest possible angle between a and b". Is there a possibility that depending on the cross product direction, the angle could be different? For example, if the cross product is in a different direction, does that affect the angle? But no, because the magnitude of the cross product is dependent only on the sine of the angle between them. The direction of the cross product is determined by the right-hand rule, but that doesn't affect the magnitude. Therefore, regardless of the direction of the cross product vector, the magnitude is fixed, so sinθ is fixed. Therefore, θ is fixed as arcsin(1/2), which gives two possibilities, but the smallest angle between them is 30 degrees.

Wait, but maybe there's a mistake here. Let me think again. Suppose that the cross product is given, but maybe there's some constraint on the vectors a and b? For example, maybe the cross product's direction is fixed, so maybe the angle is fixed? But no, the cross product's direction depends on the orientation of a and b, but the magnitude is fixed. So even if the cross product is in a particular direction, the magnitude is still 7, so sinθ is still 1/2. Therefore, θ is either 30° or 150°, so the smallest angle is 30°.

Alternatively, maybe there's a way to have a different angle? For example, if the cross product's magnitude is fixed, then sinθ is fixed, so θ is fixed up to the ambiguity of sine. But since angles between vectors are defined between 0° and 180°, so θ is either 30° or 150°, so the smallest possible angle is 30°.

Therefore, the answer should be 30 degrees.

But let me check if there's another approach. For example, using the dot product. If I can find the dot product, then I can find cosθ. But I don't have the dot product. However, maybe I can relate the cross product and dot product?

Wait, but the problem doesn't give me the dot product. However, if I can find the dot product, then I can find cosθ. But since I don't have that information, maybe I can use the relationship between cross product and dot product?

Alternatively, since I know ||a||, ||b||, and ||a × b||, maybe I can find the angle. Which is exactly what I did.

Alternatively, maybe there's a way that the cross product vector's components can give more information? For example, if I know the cross product vector, maybe I can find the angle? But no, because the cross product vector's direction is perpendicular to both a and b, but without knowing the specific directions of a and b, I can't determine the exact angle. However, the magnitude of the cross product is fixed, so the sine of the angle is fixed. Therefore, regardless of the direction of a and b, as long as their magnitudes are 2 and 7, and the cross product magnitude is 7, then sinθ is 1/2. Therefore, θ is 30° or 150°, so the smallest angle is 30°.

Therefore, I think my answer is correct.

But let me check once more. Let me compute sinθ = 1/2. So θ is 30° or 150°, so the smallest angle between them is 30°, yes. Therefore, the answer is 30 degrees.

But wait, the problem says "the smallest possible angle between a and b". Is there a possibility that depending on the cross product vector, the angle could be different? For example, if the cross product vector is in a different direction, could that affect the angle? But no, because the magnitude of the cross product is fixed. The direction of the cross product is determined by the right-hand rule, but the magnitude is fixed. Therefore, regardless of the direction of the cross product, the magnitude is fixed, so sinθ is fixed. Therefore, θ is fixed as arcsin(1/2). Therefore, the angle between them is either 30° or 150°, so the smallest possible angle is 30°.

Therefore, I think the answer is 30 degrees.

But let me check if there's a mistake in calculation. Let me recompute the magnitude of the cross product. The cross product 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. Correct. Then, ||a × b|| = 7. Then, 7 = ||a|| ||b|| sinθ => 7 = 2 * 7 * sinθ => sinθ = 7 / 14 = 1/2. Correct.

Therefore, θ = arcsin(1/2). Which is 30° or 150°, so the smallest angle is 30°.

Therefore, the answer is 30 degrees.

But just to be thorough, let me think if there's any other way this could be approached. For example, using the relationship between cross product and dot product.

We know that:

||a × b||² + (a · b)² = ||a||² ||b||²

This is the identity that relates the cross product and dot product. Let me verify that.

Yes, this is a known identity. So, if I can compute (a · b), then I can find cosθ.

But since I don't know (a · b), but I can compute it using this identity.

Let me try that.

Given that ||a × b|| = 7, ||a|| = 2, ||b|| = 7.

So, ||a × b||² = 49.

||a||² ||b||² = (2²)(7²) = 4 * 49 = 196.

Therefore, (a · b)² = ||a||² ||b||² - ||a × b||² = 196 - 49 = 147.

Therefore, a · b = ±√147.

Simplify √147: 147 = 49 * 3, so √147 = 7√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.

Therefore, 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 them is either 30° or 150°, so the smallest possible angle is 30°.

Therefore, this confirms my previous result.

So, regardless of the method, I get that the angle is 30° or 150°, so the smallest possible angle is 30 degrees.

Therefore, the answer is 30 degrees.

But let me check if there's a possibility that the cross product vector's direction could affect the sign of the dot product? For example, if the cross product is in a certain direction, does that imply something about the angle?

But no, because the cross product's direction is determined by the right-hand rule, but the magnitude is independent of direction. The dot product's sign depends on the angle between the vectors. However, since we have two possibilities for the dot product (positive or negative), which correspond to angles between 0° and 90°, or between 90° and 180°, respectively.

But since the problem asks for the smallest possible angle between a and b, regardless of the direction of the cross product, we can have both possibilities. However, the smallest angle is 30°, which is acute, whereas 150° is obtuse. Therefore, the answer is 30°.

Therefore, I think I can confidently say that the smallest possible angle 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 Magnitude of the Cross Product

The magnitude of the cross product is given by the formula:

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

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 $\theta$:

$$
\sin\theta = \frac{1}{2} \Rightarrow \theta = \arcsin\left(\frac{1}{2}\right)
$$

The arcsine of $\frac{1}{2}$ gives two possible angles in the range $[0^\circ, 180^\circ]$:

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

Since the **smallest possible angle** between two vectors is always the acute angle (i.e., the one less than or equal to $90^\circ$), the answer is:

$$
\boxed{30}
$$

---

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

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:

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

So:

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

Then:

$$
\mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \|\mathbf{b}\| \cos\theta = 14 \cos\theta \Rightarrow \cos\theta = \pm \frac{\sqrt{3}}{2}
$$

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 $30^\circ$.

---

### 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.

So, according to the formula:

||a × b|| = ||a|| ||b|| sinθ

Plugging in the known values:

7 = 2 * 7 * sinθ

Wait, let me check that again. ||a|| is 2, ||b|| is 7, so 2*7 is 14. Therefore:

7 = 14 sinθ

So, solving for sinθ:

sinθ = 7 / 14 = 1/2

Therefore, θ = arcsin(1/2). The arcsin of 1/2 is 30 degrees or 150 degrees. But since we are looking for the smallest possible angle between them, it should be 30 degrees. Wait, but hold on—is there a possibility that the angle could be something else?

Wait, the cross product's magnitude depends on sinθ, and since sinθ is positive in the first and second quadrants, θ could be between 0 and 180 degrees. But the smallest angle between two vectors is always considered to be the acute or right angle, so if θ is obtuse, the smallest angle would be 180 - θ. However, in this case, since sinθ is 1/2, θ could be 30 degrees or 150 degrees. But since the question asks for the smallest possible angle, it's 30 degrees.

But wait, let me make sure that there isn't any other consideration here. The cross product vector's direction is perpendicular to both a and b, but does that affect the angle? Hmm, no, because the magnitude of the cross product only depends on the sine of the angle between them, regardless of their direction. So, regardless of the direction of the cross product, the magnitude is determined by the sine of the angle. Therefore, the calculation seems straightforward.

But let me verify again. Let me recheck the cross product magnitude. The cross product vector is (3, 2, 6). So, the magnitude squared is 3² + 2² + 6² = 9 + 4 + 36 = 49. So, magnitude is 7. Correct. Then, ||a × b|| = 7. Then, 7 = ||a|| ||b|| sinθ => 7 = 2*7*sinθ => sinθ = 1/2. So θ is 30° or 150°, so the smallest angle is 30°. Therefore, the answer is 30 degrees.

But wait, the problem says "the smallest possible angle between a and b". Is there a possibility that depending on the cross product direction, the angle could be different? For example, if the cross product is in a different direction, does that affect the angle? But no, because the magnitude of the cross product is dependent only on the sine of the angle between them. The direction of the cross product is determined by the right-hand rule, but that doesn't affect the magnitude. Therefore, regardless of the direction of the cross product vector, the magnitude is fixed, so sinθ is fixed. Therefore, θ is fixed as arcsin(1/2), which gives two possibilities, but the smallest angle between them is 30 degrees.

Wait, but maybe there's a mistake here. Let me think again. Suppose that the cross product is given, but maybe there's some constraint on the vectors a and b? For example, maybe the cross product's direction is fixed, so maybe the angle is fixed? But no, the cross product's direction depends on the orientation of a and b, but the magnitude is fixed. So even if the cross product is in a particular direction, the magnitude is still 7, so sinθ is still 1/2. Therefore, θ is either 30° or 150°, so the smallest angle is 30°.

Alternatively, maybe there's a way to have a different angle? For example, if the cross product's magnitude is fixed, then sinθ is fixed, so θ is fixed up to the ambiguity of sine. But since angles between vectors are defined between 0° and 180°, so θ is either 30° or 150°, so the smallest possible angle is 30°.

Therefore, the answer should be 30 degrees.

But let me check if there's another approach. For example, using the dot product. If I can find the dot product, then I can find cosθ. But I don't have the dot product. However, maybe I can relate the cross product and dot product?

Wait, but the problem doesn't give me the dot product. However, if I can find the dot product, then I can find cosθ. But since I don't have that information, maybe I can use the relationship between cross product and dot product?

Alternatively, since I know ||a||, ||b||, and ||a × b||, maybe I can find the angle. Which is exactly what I did.

Alternatively, maybe there's a way that the cross product vector's components can give more information? For example, if I know the cross product vector, maybe I can find the angle? But no, because the cross product vector's direction is perpendicular to both a and b, but without knowing the specific directions of a and b, I can't determine the exact angle. However, the magnitude of the cross product is fixed, so the sine of the angle is fixed. Therefore, regardless of the direction of a and b, as long as their magnitudes are 2 and 7, and the cross product magnitude is 7, then sinθ is 1/2. Therefore, θ is 30° or 150°, so the smallest angle is 30°.

Therefore, I think my answer is correct.

But let me check once more. Let me compute sinθ = 1/2. So θ is 30° or 150°, so the smallest angle between them is 30°, yes. Therefore, the answer is 30 degrees.

But wait, the problem says "the smallest possible angle between a and b". Is there a possibility that depending on the cross product vector, the angle could be different? For example, if the cross product vector is in a different direction, could that affect the angle? But no, because the magnitude of the cross product is fixed. The direction of the cross product is determined by the right-hand rule, but the magnitude is fixed. Therefore, regardless of the direction of the cross product, the magnitude is fixed, so sinθ is fixed. Therefore, θ is fixed as arcsin(1/2). Therefore, the angle between them is either 30° or 150°, so the smallest possible angle is 30°.

Therefore, I think the answer is 30 degrees.

But let me check if there's a mistake in calculation. Let me recompute the magnitude of the cross product. The cross product 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. Correct. Then, ||a × b|| = 7. Then, 7 = ||a|| ||b|| sinθ => 7 = 2 * 7 * sinθ => sinθ = 7 / 14 = 1/2. Correct.

Therefore, θ = arcsin(1/2). Which is 30° or 150°, so the smallest angle is 30°.

Therefore, the answer is 30 degrees.

But just to be thorough, let me think if there's any other way this could be approached. For example, using the relationship between cross product and dot product.

We know that:

||a × b||² + (a · b)² = ||a||² ||b||²

This is the identity that relates the cross product and dot product. Let me verify that.

Yes, this is a known identity. So, if I can compute (a · b), then I can find cosθ.

But since I don't know (a · b), but I can compute it using this identity.

Let me try that.

Given that ||a × b|| = 7, ||a|| = 2, ||b|| = 7.

So, ||a × b||² = 49.

||a||² ||b||² = (2²)(7²) = 4 * 49 = 196.

Therefore, (a · b)² = ||a||² ||b||² - ||a × b||² = 196 - 49 = 147.

Therefore, a · b = ±√147.

Simplify √147: 147 = 49 * 3, so √147 = 7√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.

Therefore, 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 them is either 30° or 150°, so the smallest possible angle is 30°.

Therefore, this confirms my previous result.

So, regardless of the method, I get that the angle is 30° or 150°, so the smallest possible angle is 30 degrees.

Therefore, the answer is 30 degrees.

But let me check if there's a possibility that the cross product vector's direction could affect the sign of the dot product? For example, if the cross product is in a certain direction, does that imply something about the angle?

But no, because the cross product's direction is determined by the right-hand rule, but the magnitude is independent of direction. The dot product's sign depends on the angle between the vectors. However, since we have two possibilities for the dot product (positive or negative), which correspond to angles between 0° and 90°, or between 90° and 180°, respectively.

But since the problem asks for the smallest possible angle between a and b, regardless of the direction of the cross product, we can have both possibilities. However, the smallest angle is 30°, which is acute, whereas 150° is obtuse. Therefore, the answer is 30°.

Therefore, I think I can confidently say that the smallest possible angle 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 Magnitude of the Cross Product

The magnitude of the cross product is given by the formula:

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

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 $\theta$:

$$
\sin\theta = \frac{1}{2} \Rightarrow \theta = \arcsin\left(\frac{1}{2}\right)
$$

The arcsine of $\frac{1}{2}$ gives two possible angles in the range $[0^\circ, 180^\circ]$:

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

Since the **smallest possible angle** between two vectors is always the acute angle (i.e., the one less than or equal to $90^\circ$), the answer is:

$$
\boxed{30}
$$

---

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

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:

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

So:

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

Then:

$$
\mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \|\mathbf{b}\| \cos\theta = 14 \cos\theta \Rightarrow \cos\theta = \pm \frac{\sqrt{3}}{2}
$$

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 $30^\circ$.

---

### 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

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 law of cosines or something? Although since we don't have any angles, maybe that's not necessary. But just to confirm.

Alternatively, maybe considering that the third side has to be longer than the difference of the other two sides and shorter than the sum. Wait, that's essentially the triangle inequality.

The difference between 3 and 1 is 2, so the third side must be greater than 2 and less than 4. Which is the same as before. So, yeah, that gives the same result.

Therefore, the third side must be 3.

But let me check if there's a possibility of another integer. For example, if the third side was 1, but then sides would be 1, 1, 3. Let me check triangle inequality. 1 + 1 > 3? 2 > 3? No. So, invalid. Similarly, if the third side was 4, as before, 1 + 3 is not greater than 4.

So, yeah, only 3 works.

Wait, but hold on. Let me think again. If the third side is 3, then the triangle is 1, 3, 3. So, it's an isosceles triangle. That's valid.

Alternatively, if the third side was something else? But since the third side has to be integer, and between 2 and 4, only 3 is possible.

Therefore, the answer is 3.

But let me check once more. Maybe I made a mistake in the inequalities.

Suppose the sides are 1, 3, and x.

For a valid triangle, the sum of any two sides must be greater than the third.

So:

1 + 3 > x => 4 > x
1 + x > 3 => x > 2
3 + x > 1 => x > -2

So, x must be greater than 2 and less than 4. Since x is an integer, x can be 3.

Yes, that's correct.

Alternatively, if someone thought that the third side could be 2 or 4, but those are not valid because they result in degenerate triangles (where the three points are colinear). So, they are not valid triangles.

Therefore, the only possible integer value is 3.

So, I think that's the answer.

**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 for any triangle with sides $ a $, $ b $, and $ c $, the following must all be true:

$$
a + b > c \\
a + c > b \\
b + c > a
$$

Let the unknown side be $ x $. Then the sides of the triangle are $ 1 $, $ 3 $, and $ x $. Applying the triangle inequality:

---

### Step 1: Apply the inequalities

1. $ 1 + 3 > x \Rightarrow 4 > x \Rightarrow x < 4 $
2. $ 1 + x > 3 \Rightarrow x > 2 $
3. $ 3 + x > 1 \Rightarrow x > -2 $

Since side lengths are always positive, the third inequality $ x > -2 $ is automatically satisfied. So, we focus on the first two:

$$
2 < x < 4
$$

---

### Step 2: Determine integer values in the range

We are told that the third side is an **integer**. The only integer strictly between 2 and 4 is:

$$
x = 3
$$

---

### Step 3: Verify the triangle with $ x = 3 $

Let’s check if the sides $ 1 $, $ 3 $, and $ 3 $ form a valid triangle:

- $ 1 + 3 > 3 \Rightarrow 4 > 3 $ ✅
- $ 1 + 3 > 3 \Rightarrow 4 > 3 $ ✅
- $ 3 + 3 > 1 \Rightarrow 6 > 1 $ ✅

All triangle inequalities are satisfied, confirming that $ x = 3 $ is a valid solution.

---

### Step 4: Check other possible integer values

- If $ x = 2 $: $ 1 + 2 = 3 $, which is not greater than 3 → **Invalid**
- If $ x = 4 $: $ 1 + 3 = 4 $, which is not greater than 4 → **Invalid**

Thus, no other integer values satisfy the triangle inequality.

---

### 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

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 law of cosines or something? Although since we don't have any angles, maybe that's not necessary. But just to confirm.

Alternatively, maybe considering that the third side has to be longer than the difference of the other two sides and shorter than the sum. Wait, that's essentially the triangle inequality.

The difference between 3 and 1 is 2, so the third side must be greater than 2 and less than 4. Which is the same as before. So, yeah, that gives the same result.

Therefore, the third side must be 3.

But let me check if there's a possibility of another integer. For example, if the third side was 1, but then sides would be 1, 1, 3. Let me check triangle inequality. 1 + 1 > 3? 2 > 3? No. So, invalid. Similarly, if the third side was 4, as before, 1 + 3 is not greater than 4.

So, yeah, only 3 works.

Wait, but hold on. Let me think again. If the third side is 3, then the triangle is 1, 3, 3. So, it's an isosceles triangle. That's valid.

Alternatively, if the third side was something else? But since the third side has to be integer, and between 2 and 4, only 3 is possible.

Therefore, the answer is 3.

But let me check once more. Maybe I made a mistake in the inequalities.

Suppose the sides are 1, 3, and x.

For a valid triangle, the sum of any two sides must be greater than the third.

So:

1 + 3 > x => 4 > x
1 + x > 3 => x > 2
3 + x > 1 => x > -2

So, x must be greater than 2 and less than 4. Since x is an integer, x can be 3.

Yes, that's correct.

Alternatively, if someone thought that the third side could be 2 or 4, but those are not valid because they result in degenerate triangles (where the three points are colinear). So, they are not valid triangles.

Therefore, the only possible integer value is 3.

So, I think that's the answer.

**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 for any triangle with sides $ a $, $ b $, and $ c $, the following must all be true:

$$
a + b > c \\
a + c > b \\
b + c > a
$$

Let the unknown side be $ x $. Then the sides of the triangle are $ 1 $, $ 3 $, and $ x $. Applying the triangle inequality:

---

### Step 1: Apply the inequalities

1. $ 1 + 3 > x \Rightarrow 4 > x \Rightarrow x < 4 $
2. $ 1 + x > 3 \Rightarrow x > 2 $
3. $ 3 + x > 1 \Rightarrow x > -2 $

Since side lengths are always positive, the third inequality $ x > -2 $ is automatically satisfied. So, we focus on the first two:

$$
2 < x < 4
$$

---

### Step 2: Determine integer values in the range

We are told that the third side is an **integer**. The only integer strictly between 2 and 4 is:

$$
x = 3
$$

---

### Step 3: Verify the triangle with $ x = 3 $

Let’s check if the sides $ 1 $, $ 3 $, and $ 3 $ form a valid triangle:

- $ 1 + 3 > 3 \Rightarrow 4 > 3 $ ✅
- $ 1 + 3 > 3 \Rightarrow 4 > 3 $ ✅
- $ 3 + 3 > 1 \Rightarrow 6 > 1 $ ✅

All triangle inequalities are satisfied, confirming that $ x = 3 $ is a valid solution.

---

### Step 4: Check other possible integer values

- If $ x = 2 $: $ 1 + 2 = 3 $, which is not greater than 3 → **Invalid**
- If $ x = 4 $: $ 1 + 3 = 4 $, which is not greater than 4 → **Invalid**

Thus, no other integer values satisfy the triangle inequality.

---

### 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 $\boxed{3}$.