Explanation:
Okay, so I need to figure out how many different battalions can be sent to the Battle of Thermopylae based on the number of upper and lower class soldiers available. Let me start by understanding the problem.

First, the Greek army has two types of soldiers: upper class and lower class. In a certain part of Athens, there are 5 upper class soldiers and 10 lower class soldiers. The battle requires a force of 4 upper class soldiers and 8 lower class soldiers. The question is asking how many different battalions can be sent.

Hmm, so I think this is a combinatorics problem where I need to calculate the number of ways to choose the required number of soldiers from each class. Since the battle requires specific numbers of each type, I need to find the number of combinations for upper class soldiers and lower class soldiers separately and then multiply them together because the choices are independent.

Let me confirm: the problem says "different battalions," so each battalion is a group of soldiers that meets the required numbers. Since the battle demands exactly 4 upper class and 8 lower class soldiers, each battalion must consist of exactly those numbers. Therefore, the question is essentially asking how many ways we can choose 4 upper class soldiers out of 5 and 8 lower class soldiers out of 10. Then, the total number of different battalions would be the product of these two combinations.

So, first, let me recall the combination formula. The number of ways to choose k elements from a set of n elements is given by the combination formula C(n, k) = n! / (k! * (n - k)!).

Therefore, for the upper class soldiers: we have 5 soldiers and need to choose 4. So that would be C(5, 4). Let me compute that.

C(5, 4) = 5! / (4! * (5 - 4)!) = (5 * 4 * 3 * 2 * 1) / [(4 * 3 * 2 * 1) * (1)] = 5 / 1 = 5. So there are 5 ways to choose the upper class soldiers.

Now for the lower class soldiers: there are 10 soldiers and we need to choose 8. So that's C(10, 8). Let me calculate that.

C(10, 8) is the same as C(10, 2) because of the combination symmetry property C(n, k) = C(n, n - k). So C(10, 2) = 10! / (2! * 8!) = (10 * 9 * 8! ) / (2 * 1 * 8!) ) = (10 * 9)/2 = 90 / 2 = 45. Therefore, there are 45 ways to choose the lower class soldiers.

Therefore, the total number of different battalions is the product of these two numbers: 5 * 45. Let me compute that. 5 * 45 is 225.

Wait, but let me make sure I didn't make any mistakes here. Let me check the combination calculations again.

For upper class: 5 choose 4. Yes, that's 5. Because choosing 4 out of 5 is the same as leaving out 1, and there are 5 choices for who is left out. So that's correct.

For lower class: 10 choose 8. Which is 45. Let me verify with the formula. 10! / (8! * 2!) = (10 * 9 * 8! ) / (8! * 2 * 1) ) = (10 * 9)/2 = 45. Yes, that's correct. So 45 ways.

Therefore, multiplying them together: 5 * 45 = 225. So there are 225 different battalions that can be sent.

But wait, let me think again. Is there any possibility that the problem is asking for something else? For example, maybe the battalions are considered different if they have different numbers of soldiers? But no, the problem says the battle demands a force of 4 upper and 8 lower. So each battalion must have exactly 4 upper and 8 lower. Therefore, the number of different battalions is the number of ways to choose those soldiers.

Alternatively, could there be a different interpretation? For example, maybe the question is about forming battalions of any size that meet the requirement? But no, the battle demands a specific force, so the battalion must be exactly that size. Therefore, the answer is 5 * 45 = 225.

Alternatively, maybe the problem is in the translation? The original says "how many different battalions can be sent?" If "battalion" refers to a group that meets the required numbers, then yes, it's the product.

Alternatively, maybe there's a restriction that a battalion must consist of soldiers from both classes? But since the battle requires both upper and lower class soldiers, and we are choosing exactly 4 and 8, so each battalion will have both classes. So no problem there.

Alternatively, maybe the problem is considering that different orders of soldiers in the battalion count as different? But no, in combinatorics, unless specified, we assume that the order doesn't matter. So combinations are the right approach here.

Therefore, I think my reasoning is correct. The answer is 225.

But let me check once more. Suppose there are 5 upper class soldiers. Let's name them A, B, C, D, E. Choosing 4 of them: the possible combinations are all except each one individually. So, omitting A: B, C, D, E; omitting B: A, C, D, E; etc. So 5 combinations. Correct.

For lower class soldiers, 10 soldiers. Choosing 8 is equivalent to leaving out 2. The number of ways to leave out 2 soldiers from 10 is C(10, 2) = 45. So that's correct.

Therefore, total number of battalions is 5 * 45 = 225.

Therefore, I think the answer is 225.

**Final Answer**
The number of different battalions that can be sent is \boxed{225}.

To determine how many different battalions can be sent to the Battle of Thermopylae, we need to analyze the composition of the available soldiers and the specific requirements of the battle.

---

### **Step 1: Understand the Composition of the Army**

- **Upper class soldiers**: 5 available
- **Lower class soldiers**: 10 available

---

### **Step 2: Determine the Battle Requirements**

The battle requires:

- **4 upper class soldiers**
- **8 lower class soldiers**

Each battalion must consist of exactly these numbers of soldiers.

---

### **Step 3: Calculate the Number of Ways to Choose Soldiers**

We use the **combination formula** to determine how many ways we can choose the required number of soldiers from each class:

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

#### **Upper Class Soldiers:**
We need to choose 4 out of 5 upper class soldiers:

$$
C(5, 4) = \frac{5!}{4!(5-4)!} = \frac{5 \times 4!}{4! \times 1!} = 5
$$

So, there are **5** ways to choose 4 upper class soldiers.

#### **Lower Class Soldiers:**
We need to choose 8 out of 10 lower class soldiers:

$$
C(10, 8) = C(10, 2) = \frac{10!}{2!(10-2)!} = \frac{10 \times 9}{2 \times 1} = 45
$$

So, there are **45** ways to choose 8 lower class soldiers.

---

### **Step 4: Calculate the Total Number of Different Battalions**

Since the choices of upper and lower class soldiers are independent, the total number of different battalions is the product of the two combinations:

$$
\text{Total Battalions} = C(5, 4) \times C(10, 8) = 5 \times 45 = 225
$$

---

### **Final Answer**

$$
\boxed{225}
$$

[Assistant]
Explanation:
Okay, so I need to figure out how many different battalions can be sent to the Battle of Thermopylae based on the number of upper and lower class soldiers available. Let me start by understanding the problem.

First, the Greek army has two types of soldiers: upper class and lower class. In a certain part of Athens, there are 5 upper class soldiers and 10 lower class soldiers. The battle requires a force of 4 upper class soldiers and 8 lower class soldiers. The question is asking how many different battalions can be sent.

Hmm, so I think this is a combinatorics problem where I need to calculate the number of ways to choose the required number of soldiers from each class. Since the battle requires specific numbers of each type, I need to find the number of combinations for upper class soldiers and lower class soldiers separately and then multiply them together because the choices are independent.

Let me confirm: the problem says "different battalions," so each battalion is a group of soldiers that meets the required numbers. Since the battle demands exactly 4 upper class and 8 lower class soldiers, each battalion must consist of exactly those numbers. Therefore, the question is essentially asking how many ways we can choose 4 upper class soldiers out of 5 and 8 lower class soldiers out of 10. Then, the total number of different battalions would be the product of these two combinations.

So, first, let me recall the combination formula. The number of ways to choose k elements from a set of n elements is given by the combination formula C(n, k) = n! / (k! * (n - k)!).

Therefore, for the upper class soldiers: we have 5 soldiers and need to choose 4. So that would be C(5, 4). Let me compute that.

C(5, 4) = 5! / (4! * (5 - 4)!) = (5 * 4 * 3 * 2 * 1) / [(4 * 3 * 2 * 1) * (1)] = 5 / 1 = 5. So there are 5 ways to choose the upper class soldiers.

Now for the lower class soldiers: there are 10 soldiers and we need to choose 8. So that's C(10, 8). Let me calculate that.

C(10, 8) is the same as C(10, 2) because of the combination symmetry property C(n, k) = C(n, n - k). So C(10, 2) = 10! / (2! * 8!) = (10 * 9 * 8! ) / (2 * 1 * 8!) ) = (10 * 9)/2 = 90 / 2 = 45. Therefore, there are 45 ways to choose the lower class soldiers.

Therefore, the total number of different battalions is the product of these two numbers: 5 * 45. Let me compute that. 5 * 45 is 225.

Wait, but let me make sure I didn't make any mistakes here. Let me check the combination calculations again.

For upper class: 5 choose 4. Yes, that's 5. Because choosing 4 out of 5 is the same as leaving out 1, and there are 5 choices for who is left out. So that's correct.

For lower class: 10 choose 8. Which is 45. Let me verify with the formula. 10! / (8! * 2!) = (10 * 9 * 8! ) / (8! * 2 * 1) ) = (10 * 9)/2 = 45. Yes, that's correct. So 45 ways.

Therefore, multiplying them together: 5 * 45 = 225. So there are 225 different battalions that can be sent.

But wait, let me think again. Is there any possibility that the problem is asking for something else? For example, maybe the battalions are considered different if they have different numbers of soldiers? But no, the problem says the battle demands a force of 4 upper and 8 lower. So each battalion must have exactly 4 upper and 8 lower. Therefore, the number of different battalions is the number of ways to choose those soldiers.

Alternatively, could there be a different interpretation? For example, maybe the question is about forming battalions of any size that meet the requirement? But no, the battle demands a specific force, so the battalion must be exactly that size. Therefore, the answer is 5 * 45 = 225.

Alternatively, maybe the problem is in the translation? The original says "how many different battalions can be sent?" If "battalion" refers to a group that meets the required numbers, then yes, it's the product.

Alternatively, maybe there's a restriction that a battalion must consist of soldiers from both classes? But since the battle requires both upper and lower class soldiers, and we are choosing exactly 4 and 8, so each battalion will have both classes. So no problem there.

Alternatively, maybe the problem is considering that different orders of soldiers in the battalion count as different? But no, in combinatorics, unless specified, we assume that the order doesn't matter. So combinations are the right approach here.

Therefore, I think my reasoning is correct. The answer is 225.

But let me check once more. Suppose there are 5 upper class soldiers. Let's name them A, B, C, D, E. Choosing 4 of them: the possible combinations are all except each one individually. So, omitting A: B, C, D, E; omitting B: A, C, D, E; etc. So 5 combinations. Correct.

For lower class soldiers, 10 soldiers. Choosing 8 is equivalent to leaving out 2. The number of ways to leave out 2 soldiers from 10 is C(10, 2) = 45. So that's correct.

Therefore, total number of battalions is 5 * 45 = 225.

Therefore, I think the answer is 225.

**Final Answer**
The number of different battalions that can be sent is \boxed{225}.

To determine how many different battalions can be sent to the Battle of Thermopylae, we need to analyze the composition of the available soldiers and the specific requirements of the battle.

---

### **Step 1: Understand the Composition of the Army**

- **Upper class soldiers**: 5 available
- **Lower class soldiers**: 10 available

---

### **Step 2: Determine the Battle Requirements**

The battle requires:

- **4 upper class soldiers**
- **8 lower class soldiers**

Each battalion must consist of exactly these numbers of soldiers.

---

### **Step 3: Calculate the Number of Ways to Choose Soldiers**

We use the **combination formula** to determine how many ways we can choose the required number of soldiers from each class:

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

#### **Upper Class Soldiers:**
We need to choose 4 out of 5 upper class soldiers:

$$
C(5, 4) = \frac{5!}{4!(5-4)!} = \frac{5 \times 4!}{4! \times 1!} = 5
$$

So, there are **5** ways to choose 4 upper class soldiers.

#### **Lower Class Soldiers:**
We need to choose 8 out of 10 lower class soldiers:

$$
C(10, 8) = C(10, 2) = \frac{10!}{2!(10-2)!} = \frac{10 \times 9}{2 \times 1} = 45
$$

So, there are **45** ways to choose 8 lower class soldiers.

---

### **Step 4: Calculate the Total Number of Different Battalions**

Since the choices of upper and lower class soldiers are independent, the total number of different battalions is the product of the two combinations:

$$
\text{Total Battalions} = C(5, 4) \times C(10, 8) = 5 \times 45 = 225
$$

---

### **Final Answer**

$$
\boxed{225}
$$

Solution: There are $\binom{5}{4}$ different ways to choose 4 from 5 upper class soldiers. For each of these, there are $\binom{10}{8}$ ways to choose 8 lower class soldiers. The number of different battalions, then, is $\binom{5}{4}\cdot \binom{10}{8} = \boxed{225}$.


Explanation:
Okay, so I need to find the remainder when (n + 2)(n + 4)(n + 6) is divided by 7, given that n is congruent to 2 modulo 7. Let me think about how to approach this.

First, since we're dealing with modular arithmetic, maybe I can substitute n with its equivalent value modulo 7. The problem says n ≡ 2 mod 7, so that means n can be written as 7k + 2 for some integer k. But maybe instead of working with variables, I can just replace n with 2 in the expression because when we take modulo 7, the value of n is equivalent to 2. Let me check if that's valid.

Yes, in modular arithmetic, if n ≡ a mod m, then any expression involving n can be simplified by replacing n with a. So, substituting n = 2 into the expression (n + 2)(n + 4)(n + 6) should give me the same result modulo 7 as the original expression. Let me try that.

So substituting n = 2:

First term: n + 2 = 2 + 2 = 4

Second term: n + 4 = 2 + 4 = 6

Third term: n + 6 = 2 + 6 = 8

Now, multiply them together: 4 * 6 * 8. Let me compute that step by step.

First, 4 * 6 = 24. Then, 24 * 8. Let me calculate 24 * 8. Well, 20*8=160 and 4*8=32, so 160 + 32 = 192. So the product is 192. Now, I need to find the remainder when 192 is divided by 7.

Alternatively, since we're working modulo 7, maybe I can compute each term modulo 7 first and then multiply them. Let me try that approach as well to verify.

Original terms:

n + 2 ≡ 2 + 2 = 4 mod 7

n + 4 ≡ 2 + 4 = 6 mod 7

n + 6 ≡ 2 + 6 = 8 mod 7. But 8 mod 7 is 1, since 7*1=7 and 8-7=1. So that term is 1 mod 7.

Therefore, the product is 4 * 6 * 1 mod 7. Let me compute that.

First, 4 * 6 = 24. Then 24 mod 7. Let me divide 24 by 7. 7*3=21, so 24 - 21 = 3. Therefore, 24 mod 7 is 3. Then multiply by 1: 3 * 1 = 3 mod 7. So the remainder is 3.

Wait, but earlier when I calculated 192 divided by 7, let me check that. Let me divide 192 by 7. 7*27=189, so 192 - 189 = 3. So yes, 192 mod 7 is 3. So both methods give me the same result. Therefore, the remainder is 3.

But let me think again if there's another way to approach this problem, maybe using properties of modular arithmetic or something else. For example, since n ≡ 2 mod 7, then n is congruent to 2, so maybe we can think of the three terms (n + 2), (n + 4), (n + 6) as numbers that are 4, 6, 8 mod 7, which simplifies to 4, 6, 1 mod 7. Then multiplying them gives 4*6*1=24≡3 mod7.

Alternatively, maybe there's a pattern here. Let me think. If n ≡ 2 mod7, then n is 2, 9, 16, etc. Let me test with n=2. Then (2+2)(2+4)(2+6)=4*6*8=192. 192 divided by 7 is 27*7=189, remainder 3. So that's 3.

Let me try with another n that is congruent to 2 mod7. Let's take n=9. Then (9+2)=11, (9+4)=13, (9+6)=15. Multiply them: 11*13*15. Let me compute this.

First, 11*13. 10*13=130, plus 1*13=13, so 143. Then 143*15. Let me compute 143*10=1430, 143*5=715, so total is 1430 + 715 = 2145. Now divide 2145 by 7. Let me see. 7*300=2100. 2145 - 2100 = 45. 45 divided by 7 is 6*7=42, remainder 3. So again, remainder is 3.

Another example: n=16. Then (16+2)=18, (16+4)=20, (16+6)=22. Multiply them: 18*20*22. Let me compute step by step. 18*20=360. 360*22. Let me compute 360*20=7200, 360*2=720, so total is 7200 + 720 = 7920. Now divide 7920 by 7. Let me see. 7*1100=7700. 7920 - 7700 = 220. 220 divided by 7: 7*31=217, remainder 3. Again, remainder 3.

So all these examples give me remainder 3. Therefore, it seems that regardless of the value of n (as long as n ≡2 mod7), the product (n + 2)(n + 4)(n + 6) mod7 is 3.

But let me think again if there's a more algebraic way to see this. Let me consider expanding the product (n + 2)(n + 4)(n + 6). Maybe expanding it would help.

First, multiply (n + 2)(n + 4). Let me compute that:

(n + 2)(n + 4) = n^2 + 4n + 2n + 8 = n^2 + 6n + 8.

Then multiply this by (n + 6):

(n^2 + 6n + 8)(n + 6) = n^3 + 6n^2 + 6n^2 + 36n + 8n + 48.

Combine like terms:

n^3 + (6n^2 + 6n^2) + (36n + 8n) + 48 = n^3 + 12n^2 + 44n + 48.

So the expanded form is n^3 + 12n^2 + 44n + 48. Now, we need to compute this modulo 7.

Let me compute each coefficient modulo 7:

- The coefficient of n^3 is 1, which is 1 mod7.

- The coefficient of n^2 is 12. 12 divided by 7 is 1 with remainder 5, so 12 ≡ 5 mod7.

- The coefficient of n is 44. Let me divide 44 by 7: 7*6=42, so 44 - 42 = 2. Therefore, 44 ≡ 2 mod7.

- The constant term is 48. 7*6=42, 48 - 42 = 6, so 48 ≡ 6 mod7.

Therefore, the entire expression modulo7 is:

n^3 + 5n^2 + 2n + 6 mod7.

Now, since n ≡2 mod7, substitute n = 2 into this polynomial:

2^3 + 5*(2^2) + 2*(2) + 6.

Compute each term:

2^3 = 8

5*(2^2) = 5*4 = 20

2*(2) = 4

Constant term is 6.

Adding them up: 8 + 20 + 4 + 6 = 38.

Now compute 38 mod7. 7*5=35, so 38 - 35 = 3. Therefore, 38 ≡3 mod7.

So again, we get 3. Therefore, regardless of the method, the remainder is 3.

Alternatively, maybe there's a smarter way using properties of numbers. Let me think. Since n ≡2 mod7, then n is congruent to 2, so the numbers n+2, n+4, n+6 are congruent to 4, 6, 1 mod7. So the product is 4*6*1 mod7.

But 4*6 is 24, which is 3 mod7, and 3*1 is 3 mod7. So that's straightforward.

Alternatively, maybe think about the three numbers n+2, n+4, n+6. Since n ≡2 mod7, then n+2 ≡4, n+4≡6, n+6≡1. So they are 4, 6, 1.

Alternatively, note that 4, 6, 1 are numbers that are 4, 6, and 1. But 4*6*1=24≡3 mod7.

Alternatively, maybe think about the product (n + 2)(n + 4)(n + 6) as something related to (n + 2)(n + 4)(n + 6). Since n ≡2 mod7, then n is 2 mod7, so maybe we can think of these terms as numbers in the form of 2 + 2, 2 + 4, 2 + 6. Which is 4, 6, 8. Which is 4, 6, 1 mod7.

Alternatively, maybe think of the product as (n + 2)(n + 4)(n + 6) = (n + 2)(n + 4)(n + 6). Since n ≡2 mod7, then n + 2 ≡4, n + 4≡6, n + 6≡1. So the product is 4*6*1=24≡3 mod7.

Alternatively, maybe think of the product as (n + 2)(n + 4)(n + 6) mod7. Since n ≡2, then substituting n=2 gives the same result.

But maybe there's another way. Let me think about the fact that 2, 4, 6 are even numbers. But not sure if that helps.

Alternatively, think about the fact that (n + 2)(n + 4)(n + 6) is the product of three consecutive even numbers? Wait, no. If n is 2 mod7, then n is even? Wait, n ≡2 mod7. But 2 is even, but n could be even or odd depending on the value. Wait, no. If n ≡2 mod7, then n can be 2, 9, 16, etc. So 2 is even, 9 is odd, 16 is even, etc. So n can be even or odd. Therefore, the terms n + 2, n + 4, n + 6 can be even or odd. But maybe that's not helpful.

Alternatively, think about the product modulo 7. Since 7 is prime, maybe there's some properties. But since we already have the answer through substitution, maybe that's sufficient.

But let me check if there's another approach. For example, since n ≡2 mod7, then n = 7k + 2. Then, substituting into the product:

(n + 2)(n + 4)(n + 6) = (7k + 2 + 2)(7k + 2 + 4)(7k + 2 + 6) = (7k + 4)(7k + 6)(7k + 8).

Now, expanding this product would give terms with 7k, but when we take modulo7, all terms with 7k will be 0 mod7. Therefore, the entire product mod7 is equal to (4)(6)(8) mod7. Which is the same as before. 4*6*8 mod7. Which is 4*6*1 mod7, since 8≡1 mod7. So again, 4*6*1=24≡3 mod7.

Therefore, regardless of the method, the answer is 3.

Alternatively, maybe think of the product (n + 2)(n + 4)(n + 6) as (n + 2)(n + 4)(n + 6) mod7. Since n ≡2, then n + 2 ≡4, n + 4≡6, n + 6≡1. So the product is 4*6*1=24≡3 mod7.

Therefore, all methods lead to the same conclusion.

But just to make sure, let me try another example. Let me take n= -5. Wait, n ≡2 mod7. Let me check if -5 ≡2 mod7. -5 +7=2, so yes, -5 ≡2 mod7. So n=-5. Then compute (n + 2)(n + 4)(n + 6).

n + 2 = -5 + 2 = -3

n + 4 = -5 + 4 = -1

n + 6 = -5 + 6 = 1

Multiply them: (-3)*(-1)*(1) = 3*1=3. Then 3 mod7 is 3. So again, remainder is 3.

Another example: n= -12. Since -12 mod7: 7*(-2)=-14, so -12 - (-14)=2. So -12 ≡2 mod7. Then compute (n + 2)(n + 4)(n + 6):

n + 2 = -12 + 2 = -10

n + 4 = -12 + 4 = -8

n + 6 = -12 + 6 = -6

Multiply them: (-10)*(-8)*(-6). Let me compute step by step. First, (-10)*(-8)=80. Then 80*(-6)= -480. Now compute -480 mod7.

First, find 480 divided by 7. Let me compute 7*68=476. 480 - 476=4. Therefore, 480 ≡4 mod7. Therefore, -480 ≡ -4 mod7. Which is equivalent to 3 mod7 because -4 +7=3. Therefore, the remainder is 3.

So again, same result.

Therefore, regardless of the value of n, as long as n ≡2 mod7, the product (n + 2)(n + 4)(n + 6) mod7 is 3.

Therefore, the answer is 3.

But just to recap, the key idea here is that when dealing with modular arithmetic, we can substitute the congruence value of n into the expression. Since n ≡2 mod7, replacing n with 2 simplifies the problem, and then computing the product modulo7 gives the answer. Alternatively, expanding the polynomial and reducing each coefficient modulo7 also leads to the same result. Testing with multiple examples confirms that the remainder is consistently 3.

Therefore, I can be confident that the remainder is 3 when (n + 2)(n + 4)(n + 6) is divided by 7, given that n ≡2 mod7.

**Final Answer**
The remainder is \boxed{3}.

To solve the problem, we are given that:

$$
n \equiv 2 \pmod{7}
$$

We are asked to find the remainder when the expression:

$$
(n + 2)(n + 4)(n + 6)
$$

is divided by 7.

---

### Step 1: Use the Given Congruence

Since $ n \equiv 2 \pmod{7} $, we can replace $ n $ with 2 in the expression. This is valid because in modular arithmetic, if $ n \equiv a \pmod{m} $, then any expression involving $ n $ can be simplified by replacing $ n $ with $ a $.

So, we compute:

$$
(n + 2)(n + 4)(n + 6) \equiv (2 + 2)(2 + 4)(2 + 6) \pmod{7}
$$

$$
\equiv (4)(6)(8) \pmod{7}
$$

---

### Step 2: Simplify Each Term Modulo 7

Now, reduce each term modulo 7:

- $ 4 \mod 7 = 4 $
- $ 6 \mod 7 = 6 $
- $ 8 \mod 7 = 1 $

So the expression becomes:

$$
4 \cdot 6 \cdot 1 \pmod{7}
$$

---

### Step 3: Compute the Product Modulo 7

First, compute $ 4 \cdot 6 = 24 $, then:

$$
24 \mod 7 = 3
$$

So the final result is:

$$
3 \pmod{7}
$$

---

### Final Answer

$$
\boxed{3}
$$

[Assistant]
Explanation:
Okay, so I need to find the remainder when (n + 2)(n + 4)(n + 6) is divided by 7, given that n is congruent to 2 modulo 7. Let me think about how to approach this.

First, since we're dealing with modular arithmetic, maybe I can substitute n with its equivalent value modulo 7. The problem says n ≡ 2 mod 7, so that means n can be written as 7k + 2 for some integer k. But maybe instead of working with variables, I can just replace n with 2 in the expression because when we take modulo 7, the value of n is equivalent to 2. Let me check if that's valid.

Yes, in modular arithmetic, if n ≡ a mod m, then any expression involving n can be simplified by replacing n with a. So, substituting n = 2 into the expression (n + 2)(n + 4)(n + 6) should give me the same result modulo 7 as the original expression. Let me try that.

So substituting n = 2:

First term: n + 2 = 2 + 2 = 4

Second term: n + 4 = 2 + 4 = 6

Third term: n + 6 = 2 + 6 = 8

Now, multiply them together: 4 * 6 * 8. Let me compute that step by step.

First, 4 * 6 = 24. Then, 24 * 8. Let me calculate 24 * 8. Well, 20*8=160 and 4*8=32, so 160 + 32 = 192. So the product is 192. Now, I need to find the remainder when 192 is divided by 7.

Alternatively, since we're working modulo 7, maybe I can compute each term modulo 7 first and then multiply them. Let me try that approach as well to verify.

Original terms:

n + 2 ≡ 2 + 2 = 4 mod 7

n + 4 ≡ 2 + 4 = 6 mod 7

n + 6 ≡ 2 + 6 = 8 mod 7. But 8 mod 7 is 1, since 7*1=7 and 8-7=1. So that term is 1 mod 7.

Therefore, the product is 4 * 6 * 1 mod 7. Let me compute that.

First, 4 * 6 = 24. Then 24 mod 7. Let me divide 24 by 7. 7*3=21, so 24 - 21 = 3. Therefore, 24 mod 7 is 3. Then multiply by 1: 3 * 1 = 3 mod 7. So the remainder is 3.

Wait, but earlier when I calculated 192 divided by 7, let me check that. Let me divide 192 by 7. 7*27=189, so 192 - 189 = 3. So yes, 192 mod 7 is 3. So both methods give me the same result. Therefore, the remainder is 3.

But let me think again if there's another way to approach this problem, maybe using properties of modular arithmetic or something else. For example, since n ≡ 2 mod 7, then n is congruent to 2, so maybe we can think of the three terms (n + 2), (n + 4), (n + 6) as numbers that are 4, 6, 8 mod 7, which simplifies to 4, 6, 1 mod 7. Then multiplying them gives 4*6*1=24≡3 mod7.

Alternatively, maybe there's a pattern here. Let me think. If n ≡ 2 mod7, then n is 2, 9, 16, etc. Let me test with n=2. Then (2+2)(2+4)(2+6)=4*6*8=192. 192 divided by 7 is 27*7=189, remainder 3. So that's 3.

Let me try with another n that is congruent to 2 mod7. Let's take n=9. Then (9+2)=11, (9+4)=13, (9+6)=15. Multiply them: 11*13*15. Let me compute this.

First, 11*13. 10*13=130, plus 1*13=13, so 143. Then 143*15. Let me compute 143*10=1430, 143*5=715, so total is 1430 + 715 = 2145. Now divide 2145 by 7. Let me see. 7*300=2100. 2145 - 2100 = 45. 45 divided by 7 is 6*7=42, remainder 3. So again, remainder is 3.

Another example: n=16. Then (16+2)=18, (16+4)=20, (16+6)=22. Multiply them: 18*20*22. Let me compute step by step. 18*20=360. 360*22. Let me compute 360*20=7200, 360*2=720, so total is 7200 + 720 = 7920. Now divide 7920 by 7. Let me see. 7*1100=7700. 7920 - 7700 = 220. 220 divided by 7: 7*31=217, remainder 3. Again, remainder 3.

So all these examples give me remainder 3. Therefore, it seems that regardless of the value of n (as long as n ≡2 mod7), the product (n + 2)(n + 4)(n + 6) mod7 is 3.

But let me think again if there's a more algebraic way to see this. Let me consider expanding the product (n + 2)(n + 4)(n + 6). Maybe expanding it would help.

First, multiply (n + 2)(n + 4). Let me compute that:

(n + 2)(n + 4) = n^2 + 4n + 2n + 8 = n^2 + 6n + 8.

Then multiply this by (n + 6):

(n^2 + 6n + 8)(n + 6) = n^3 + 6n^2 + 6n^2 + 36n + 8n + 48.

Combine like terms:

n^3 + (6n^2 + 6n^2) + (36n + 8n) + 48 = n^3 + 12n^2 + 44n + 48.

So the expanded form is n^3 + 12n^2 + 44n + 48. Now, we need to compute this modulo 7.

Let me compute each coefficient modulo 7:

- The coefficient of n^3 is 1, which is 1 mod7.

- The coefficient of n^2 is 12. 12 divided by 7 is 1 with remainder 5, so 12 ≡ 5 mod7.

- The coefficient of n is 44. Let me divide 44 by 7: 7*6=42, so 44 - 42 = 2. Therefore, 44 ≡ 2 mod7.

- The constant term is 48. 7*6=42, 48 - 42 = 6, so 48 ≡ 6 mod7.

Therefore, the entire expression modulo7 is:

n^3 + 5n^2 + 2n + 6 mod7.

Now, since n ≡2 mod7, substitute n = 2 into this polynomial:

2^3 + 5*(2^2) + 2*(2) + 6.

Compute each term:

2^3 = 8

5*(2^2) = 5*4 = 20

2*(2) = 4

Constant term is 6.

Adding them up: 8 + 20 + 4 + 6 = 38.

Now compute 38 mod7. 7*5=35, so 38 - 35 = 3. Therefore, 38 ≡3 mod7.

So again, we get 3. Therefore, regardless of the method, the remainder is 3.

Alternatively, maybe there's a smarter way using properties of numbers. Let me think. Since n ≡2 mod7, then n is congruent to 2, so the numbers n+2, n+4, n+6 are congruent to 4, 6, 1 mod7. So the product is 4*6*1 mod7.

But 4*6 is 24, which is 3 mod7, and 3*1 is 3 mod7. So that's straightforward.

Alternatively, maybe think about the three numbers n+2, n+4, n+6. Since n ≡2 mod7, then n+2 ≡4, n+4≡6, n+6≡1. So they are 4, 6, 1.

Alternatively, note that 4, 6, 1 are numbers that are 4, 6, and 1. But 4*6*1=24≡3 mod7.

Alternatively, maybe think about the product (n + 2)(n + 4)(n + 6) as something related to (n + 2)(n + 4)(n + 6). Since n ≡2 mod7, then n is 2 mod7, so maybe we can think of these terms as numbers in the form of 2 + 2, 2 + 4, 2 + 6. Which is 4, 6, 8. Which is 4, 6, 1 mod7.

Alternatively, maybe think of the product as (n + 2)(n + 4)(n + 6) = (n + 2)(n + 4)(n + 6). Since n ≡2 mod7, then n + 2 ≡4, n + 4≡6, n + 6≡1. So the product is 4*6*1=24≡3 mod7.

Alternatively, maybe think of the product as (n + 2)(n + 4)(n + 6) mod7. Since n ≡2, then substituting n=2 gives the same result.

But maybe there's another way. Let me think about the fact that 2, 4, 6 are even numbers. But not sure if that helps.

Alternatively, think about the fact that (n + 2)(n + 4)(n + 6) is the product of three consecutive even numbers? Wait, no. If n is 2 mod7, then n is even? Wait, n ≡2 mod7. But 2 is even, but n could be even or odd depending on the value. Wait, no. If n ≡2 mod7, then n can be 2, 9, 16, etc. So 2 is even, 9 is odd, 16 is even, etc. So n can be even or odd. Therefore, the terms n + 2, n + 4, n + 6 can be even or odd. But maybe that's not helpful.

Alternatively, think about the product modulo 7. Since 7 is prime, maybe there's some properties. But since we already have the answer through substitution, maybe that's sufficient.

But let me check if there's another approach. For example, since n ≡2 mod7, then n = 7k + 2. Then, substituting into the product:

(n + 2)(n + 4)(n + 6) = (7k + 2 + 2)(7k + 2 + 4)(7k + 2 + 6) = (7k + 4)(7k + 6)(7k + 8).

Now, expanding this product would give terms with 7k, but when we take modulo7, all terms with 7k will be 0 mod7. Therefore, the entire product mod7 is equal to (4)(6)(8) mod7. Which is the same as before. 4*6*8 mod7. Which is 4*6*1 mod7, since 8≡1 mod7. So again, 4*6*1=24≡3 mod7.

Therefore, regardless of the method, the answer is 3.

Alternatively, maybe think of the product (n + 2)(n + 4)(n + 6) as (n + 2)(n + 4)(n + 6) mod7. Since n ≡2, then n + 2 ≡4, n + 4≡6, n + 6≡1. So the product is 4*6*1=24≡3 mod7.

Therefore, all methods lead to the same conclusion.

But just to make sure, let me try another example. Let me take n= -5. Wait, n ≡2 mod7. Let me check if -5 ≡2 mod7. -5 +7=2, so yes, -5 ≡2 mod7. So n=-5. Then compute (n + 2)(n + 4)(n + 6).

n + 2 = -5 + 2 = -3

n + 4 = -5 + 4 = -1

n + 6 = -5 + 6 = 1

Multiply them: (-3)*(-1)*(1) = 3*1=3. Then 3 mod7 is 3. So again, remainder is 3.

Another example: n= -12. Since -12 mod7: 7*(-2)=-14, so -12 - (-14)=2. So -12 ≡2 mod7. Then compute (n + 2)(n + 4)(n + 6):

n + 2 = -12 + 2 = -10

n + 4 = -12 + 4 = -8

n + 6 = -12 + 6 = -6

Multiply them: (-10)*(-8)*(-6). Let me compute step by step. First, (-10)*(-8)=80. Then 80*(-6)= -480. Now compute -480 mod7.

First, find 480 divided by 7. Let me compute 7*68=476. 480 - 476=4. Therefore, 480 ≡4 mod7. Therefore, -480 ≡ -4 mod7. Which is equivalent to 3 mod7 because -4 +7=3. Therefore, the remainder is 3.

So again, same result.

Therefore, regardless of the value of n, as long as n ≡2 mod7, the product (n + 2)(n + 4)(n + 6) mod7 is 3.

Therefore, the answer is 3.

But just to recap, the key idea here is that when dealing with modular arithmetic, we can substitute the congruence value of n into the expression. Since n ≡2 mod7, replacing n with 2 simplifies the problem, and then computing the product modulo7 gives the answer. Alternatively, expanding the polynomial and reducing each coefficient modulo7 also leads to the same result. Testing with multiple examples confirms that the remainder is consistently 3.

Therefore, I can be confident that the remainder is 3 when (n + 2)(n + 4)(n + 6) is divided by 7, given that n ≡2 mod7.

**Final Answer**
The remainder is \boxed{3}.

To solve the problem, we are given that:

$$
n \equiv 2 \pmod{7}
$$

We are asked to find the remainder when the expression:

$$
(n + 2)(n + 4)(n + 6)
$$

is divided by 7.

---

### Step 1: Use the Given Congruence

Since $ n \equiv 2 \pmod{7} $, we can replace $ n $ with 2 in the expression. This is valid because in modular arithmetic, if $ n \equiv a \pmod{m} $, then any expression involving $ n $ can be simplified by replacing $ n $ with $ a $.

So, we compute:

$$
(n + 2)(n + 4)(n + 6) \equiv (2 + 2)(2 + 4)(2 + 6) \pmod{7}
$$

$$
\equiv (4)(6)(8) \pmod{7}
$$

---

### Step 2: Simplify Each Term Modulo 7

Now, reduce each term modulo 7:

- $ 4 \mod 7 = 4 $
- $ 6 \mod 7 = 6 $
- $ 8 \mod 7 = 1 $

So the expression becomes:

$$
4 \cdot 6 \cdot 1 \pmod{7}
$$

---

### Step 3: Compute the Product Modulo 7

First, compute $ 4 \cdot 6 = 24 $, then:

$$
24 \mod 7 = 3
$$

So the final result is:

$$
3 \pmod{7}
$$

---

### Final Answer

$$
\boxed{3}
$$

Solution: If $n \equiv 2 \pmod{7}$, then $(n + 2)(n + 4)(n + 6) \equiv 4 \cdot 6 \cdot 8 \equiv 4 \cdot 6 \cdot 1 \equiv 24 \equiv \boxed{3} \pmod{7}$.


Explanation:
Okay, so I need to find the roots of the equation $(x - 3)^3 + (x - 7)^3 = (2x - 10)^3$. Hmm, let me think about how to approach this.

First, maybe I should expand all the terms and simplify the equation. Let me start by expanding the left side.

The left side is $(x - 3)^3 + (x - 7)^3$. I remember that $(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$, so maybe I can apply that formula here. Let me do that for each term.

Starting with $(x - 3)^3$:

$$
(x - 3)^3 = x^3 - 3x^2(3) + 3x(3)^2 - 3^3 = x^3 - 9x^2 + 27x - 27
$$

Wait, let me check that again. The formula is $a^3 - 3a^2b + 3ab^2 - b^3$, so substituting $a = x$ and $b = 3$:

$$
x^3 - 3x^2(3) + 3x(3)^2 - (3)^3 = x^3 - 9x^2 + 27x - 27
$$

Yes, that seems right. Now for $(x - 7)^3$:

$$
(x - 7)^3 = x^3 - 3x^2(7) + 3x(7)^2 - 7^3 = x^3 - 21x^2 + 147x - 343
$$

Let me verify that again. Using the same formula with $a = x$ and $b = 7$:

$$
x^3 - 3x^2(7) + 3x(7)^2 - 7^3 = x^3 - 21x^2 + 147x - 343
$$

Okay, that looks correct. So adding the two expanded terms together:

Left side becomes:

$$
(x^3 - 9x^2 + 27x - 27) + (x^3 - 21x^2 + 147x - 343)
$$

Combine like terms:

- $x^3 + x^3 = 2x^3$
- $-9x^2 -21x^2 = -30x^2$
- $27x + 147x = 174x$
- $-27 - 343 = -370$

So the left side simplifies to:

$$
2x^3 - 30x^2 + 174x - 370
$$

Now, let me expand the right side, which is $(2x - 10)^3$. Again, using the same formula. Let me recall that $(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$. Here, $a = 2x$ and $b = 10$:

$$
(2x)^3 - 3(2x)^2(10) + 3(2x)(10)^2 - (10)^3
$$

Calculating each term:

- $(2x)^3 = 8x^3$
- $-3(2x)^2(10) = -3*(4x^2)*10 = -120x^2$
- $3(2x)(10)^2 = 3*2x*100 = 600x$
- $-(10)^3 = -1000$

So putting it all together:

$$
8x^3 - 120x^2 + 600x - 1000
$$

Therefore, the original equation becomes:

Left side: $2x^3 - 30x^2 + 174x - 370$

Right side: $8x^3 - 120x^2 + 600x - 1000$

So, bringing all terms to one side:

Left side minus right side equals zero:

$$
(2x^3 - 30x^2 + 174x - 370) - (8x^3 - 120x^2 + 600x - 1000) = 0
$$

Let me compute each term:

- $2x^3 - 8x^3 = -6x^3$
- $-30x^2 - (-120x^2) = -30x^2 + 120x^2 = 90x^2$
- $174x - 600x = -426x$
- $-370 - (-1000) = -370 + 1000 = 630$

So the equation simplifies to:

$$
-6x^3 + 90x^2 - 426x + 630 = 0
$$

Hmm, maybe I can factor out a common factor here. Let me check if all coefficients are divisible by something. Let's see:

-6, 90, -426, 630.

Looking for a common divisor. Let's check 6:

-6 is divisible by 6, 90 is divisible by 6, 426 divided by 6 is 71, and 630 divided by 6 is 105. Wait, 426 divided by 6 is 71? Let me check:

6*70 = 420, so 6*71 = 426. Yes. And 6*105 = 630. So yes, all coefficients are divisible by 6. Let me factor that out:

$$
6(-x^3 + 15x^2 - 71x + 105) = 0
$$

So, dividing both sides by 6:

$$
 -x^3 + 15x^2 - 71x + 105 = 0
$$

Alternatively, multiplying both sides by -1 to make the leading coefficient positive:

$$
x^3 - 15x^2 + 71x - 105 = 0
$$

So now, we have a cubic equation: $x^3 - 15x^2 + 71x - 105 = 0$. Let me try to factor this. Maybe there are rational roots. By Rational Root Theorem, possible rational roots are factors of 105 over factors of 1, so ±1, ±3, ±5, ±7, ±15, ±21, ±35, ±105.

Let me test x=1:

1 -15 +71 -105 = 1 -15 is -14, -14 +71 is 57, 57 -105 is -48 ≠ 0.

x=3:

27 - 15*9 + 71*3 -105 = 27 -135 + 213 -105. Let's compute step by step:

27 -135 = -108; -108 +213 = 105; 105 -105 = 0. Oh! x=3 is a root!

So, (x - 3) is a factor. Let's perform polynomial division or use synthetic division to factor it out.

Let me use synthetic division with root 3:

Coefficients of the cubic: 1 (x^3), -15 (x^2), 71 (x), -105 (constant).

Set up synthetic division:

3 | 1   -15   71   -105
  |     3   -36   105
  ---------------------
    1   -12   35     0

So, the cubic factors as (x - 3)(x^2 -12x +35). Now, let's factor the quadratic: x^2 -12x +35.

Looking for two numbers that multiply to 35 and add to -12. Let me think: factors of 35 are 1 & 35, 5 &7. 5 and 7 add to 12. So since the middle term is -12x, both numbers should be negative: -5 and -7.

Therefore, x^2 -12x +35 factors as (x -5)(x -7).

So, putting it all together, the cubic factors as:

(x - 3)(x -5)(x -7) = 0

Therefore, the roots are x = 3, x = 5, x = 7.

Wait, but hold on. Let me check if these roots satisfy the original equation. Because sometimes when we manipulate equations, especially expanding and moving terms around, we might introduce extraneous roots or maybe lose some? Although in this case, since we just moved terms around and factored, I think they should be valid. But let me check.

First, check x=3:

Left side: (3 -3)^3 + (3 -7)^3 = 0 + (-4)^3 = -64

Right side: (2*3 -10)^3 = (6 -10)^3 = (-4)^3 = -64. So yes, equal. Good.

x=5:

Left side: (5 -3)^3 + (5 -7)^3 = (2)^3 + (-2)^3 = 8 -8 = 0

Right side: (2*5 -10)^3 = (10 -10)^3 = 0^3 = 0. Equal. Good.

x=7:

Left side: (7 -3)^3 + (7 -7)^3 = 4^3 + 0 = 64 + 0 = 64

Right side: (2*7 -10)^3 = (14 -10)^3 = 4^3 = 64. Equal. Perfect.

So all three roots are valid. Therefore, the solutions are x=3, x=5, x=7.

But wait, let me think again. The original equation was $(x - 3)^3 + (x - 7)^3 = (2x - 10)^3$. When I expanded both sides, moved everything to left, and simplified, I ended up with a cubic equation which had roots 3,5,7. But maybe there's another way to approach this problem that could have made it easier?

Alternatively, maybe there's a substitution or identity that could have been used. Let me think. For example, sometimes equations of the form a^3 + b^3 = c^3 can be approached using identities. But in this case, the left side is a sum of two cubes, and the right side is another cube. However, the standard identity for a^3 + b^3 is (a + b)(a^2 - ab + b^2), but I don't know if that helps here. Let me try.

Let me denote a = x - 3 and b = x - 7. Then the left side is a^3 + b^3. The right side is (2x -10)^3. Let me see if there's a relationship between a, b, and 2x -10.

Note that 2x -10 = 2(x -5). Also, note that a = x -3 and b = x -7. Let me compute a + b:

a + b = (x -3) + (x -7) = 2x -10. Oh! That's exactly the right side. So, the equation becomes:

a^3 + b^3 = (a + b)^3.

So, substituting back:

a^3 + b^3 = (a + b)^3.

But we know that (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3. Therefore, subtracting a^3 + b^3 from both sides:

0 = 3a^2b + 3ab^2

Which simplifies to:

0 = 3ab(a + b)

Therefore, the equation reduces to 3ab(a + b) = 0. Therefore, either a = 0, b = 0, or a + b = 0.

So, solving each case:

Case 1: a = 0 => x - 3 = 0 => x = 3

Case 2: b = 0 => x -7 = 0 => x =7

Case 3: a + b = 0 => (x -3) + (x -7) = 0 => 2x -10 = 0 => x = 5

Therefore, the solutions are x=3, x=5, x=7. Which matches the previous result.

This method seems much more straightforward. So, by recognizing that the right side is (a + b)^3 where a and b are the terms on the left, we can use the identity for the sum of cubes and simplify the equation significantly.

This shows that sometimes recognizing algebraic identities can make solving equations much easier than expanding everything out.

But since I already solved it both ways, and both methods give the same roots, I can be confident that the roots are indeed 3, 5, and 7.

Therefore, the answer is x = 3, 5, 7.

**Final Answer**
The roots are \boxed{3}, \boxed{5}, and \boxed{7}.

To solve the equation:

$$
(x - 3)^3 + (x - 7)^3 = (2x - 10)^3
$$

we begin by recognizing a key algebraic identity. Let us define:

- $ a = x - 3 $
- $ b = x - 7 $

Then the left-hand side becomes:

$$
a^3 + b^3
$$

and the right-hand side becomes:

$$
(2x - 10)^3 = (a + b)^3
$$

So the equation becomes:

$$
a^3 + b^3 = (a + b)^3
$$

We now use the identity for the cube of a sum:

$$
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
$$

Subtracting $ a^3 + b^3 $ from both sides:

$$
0 = 3a^2b + 3ab^2
$$

Factor out $ 3ab $:

$$
0 = 3ab(a + b)
$$

This gives us three possible cases:

1. $ a = 0 \Rightarrow x - 3 = 0 \Rightarrow x = 3 $
2. $ b = 0 \Rightarrow x - 7 = 0 \Rightarrow x = 7 $
3. $ a + b = 0 \Rightarrow (x - 3) + (x - 7) = 0 \Rightarrow 2x - 10 = 0 \Rightarrow x = 5 $

---

### Verification of Roots

Let’s verify each root in the original equation:

- **For $ x = 3 $:**
  $$
  (3 - 3)^3 + (3 - 7)^3 = 0 + (-4)^3 = -64
  $$
  $$
  (2 \cdot 3 - 10)^3 = (6 - 10)^3 = (-4)^3 = -64
  $$
  ✅ Equal

- **For $ x = 5 $:**
  $$
  (5 - 3)^3 + (5 - 7)^3 = 2^3 + (-2)^3 = 8 - 8 = 0
  $$
  $$
  (2 \cdot 5 - 10)^3 = (10 - 10)^3 = 0^3 = 0
  $$
  ✅ Equal

- **For $ x = 7 $:**
  $$
  (7 - 3)^3 + (7 - 7)^3 = 4^3 + 0 = 64
  $$
  $$
  (2 \cdot 7 - 10)^3 = (14 - 10)^3 = 4^3 = 64
  $$
  ✅ Equal

---

### Final Answer

The roots of the equation are:

$$
\boxed{3}, \boxed{5}, \boxed{7}
$$

[Assistant]
Explanation:
Okay, so I need to find the roots of the equation $(x - 3)^3 + (x - 7)^3 = (2x - 10)^3$. Hmm, let me think about how to approach this.

First, maybe I should expand all the terms and simplify the equation. Let me start by expanding the left side.

The left side is $(x - 3)^3 + (x - 7)^3$. I remember that $(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$, so maybe I can apply that formula here. Let me do that for each term.

Starting with $(x - 3)^3$:

$$
(x - 3)^3 = x^3 - 3x^2(3) + 3x(3)^2 - 3^3 = x^3 - 9x^2 + 27x - 27
$$

Wait, let me check that again. The formula is $a^3 - 3a^2b + 3ab^2 - b^3$, so substituting $a = x$ and $b = 3$:

$$
x^3 - 3x^2(3) + 3x(3)^2 - (3)^3 = x^3 - 9x^2 + 27x - 27
$$

Yes, that seems right. Now for $(x - 7)^3$:

$$
(x - 7)^3 = x^3 - 3x^2(7) + 3x(7)^2 - 7^3 = x^3 - 21x^2 + 147x - 343
$$

Let me verify that again. Using the same formula with $a = x$ and $b = 7$:

$$
x^3 - 3x^2(7) + 3x(7)^2 - 7^3 = x^3 - 21x^2 + 147x - 343
$$

Okay, that looks correct. So adding the two expanded terms together:

Left side becomes:

$$
(x^3 - 9x^2 + 27x - 27) + (x^3 - 21x^2 + 147x - 343)
$$

Combine like terms:

- $x^3 + x^3 = 2x^3$
- $-9x^2 -21x^2 = -30x^2$
- $27x + 147x = 174x$
- $-27 - 343 = -370$

So the left side simplifies to:

$$
2x^3 - 30x^2 + 174x - 370
$$

Now, let me expand the right side, which is $(2x - 10)^3$. Again, using the same formula. Let me recall that $(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$. Here, $a = 2x$ and $b = 10$:

$$
(2x)^3 - 3(2x)^2(10) + 3(2x)(10)^2 - (10)^3
$$

Calculating each term:

- $(2x)^3 = 8x^3$
- $-3(2x)^2(10) = -3*(4x^2)*10 = -120x^2$
- $3(2x)(10)^2 = 3*2x*100 = 600x$
- $-(10)^3 = -1000$

So putting it all together:

$$
8x^3 - 120x^2 + 600x - 1000
$$

Therefore, the original equation becomes:

Left side: $2x^3 - 30x^2 + 174x - 370$

Right side: $8x^3 - 120x^2 + 600x - 1000$

So, bringing all terms to one side:

Left side minus right side equals zero:

$$
(2x^3 - 30x^2 + 174x - 370) - (8x^3 - 120x^2 + 600x - 1000) = 0
$$

Let me compute each term:

- $2x^3 - 8x^3 = -6x^3$
- $-30x^2 - (-120x^2) = -30x^2 + 120x^2 = 90x^2$
- $174x - 600x = -426x$
- $-370 - (-1000) = -370 + 1000 = 630$

So the equation simplifies to:

$$
-6x^3 + 90x^2 - 426x + 630 = 0
$$

Hmm, maybe I can factor out a common factor here. Let me check if all coefficients are divisible by something. Let's see:

-6, 90, -426, 630.

Looking for a common divisor. Let's check 6:

-6 is divisible by 6, 90 is divisible by 6, 426 divided by 6 is 71, and 630 divided by 6 is 105. Wait, 426 divided by 6 is 71? Let me check:

6*70 = 420, so 6*71 = 426. Yes. And 6*105 = 630. So yes, all coefficients are divisible by 6. Let me factor that out:

$$
6(-x^3 + 15x^2 - 71x + 105) = 0
$$

So, dividing both sides by 6:

$$
 -x^3 + 15x^2 - 71x + 105 = 0
$$

Alternatively, multiplying both sides by -1 to make the leading coefficient positive:

$$
x^3 - 15x^2 + 71x - 105 = 0
$$

So now, we have a cubic equation: $x^3 - 15x^2 + 71x - 105 = 0$. Let me try to factor this. Maybe there are rational roots. By Rational Root Theorem, possible rational roots are factors of 105 over factors of 1, so ±1, ±3, ±5, ±7, ±15, ±21, ±35, ±105.

Let me test x=1:

1 -15 +71 -105 = 1 -15 is -14, -14 +71 is 57, 57 -105 is -48 ≠ 0.

x=3:

27 - 15*9 + 71*3 -105 = 27 -135 + 213 -105. Let's compute step by step:

27 -135 = -108; -108 +213 = 105; 105 -105 = 0. Oh! x=3 is a root!

So, (x - 3) is a factor. Let's perform polynomial division or use synthetic division to factor it out.

Let me use synthetic division with root 3:

Coefficients of the cubic: 1 (x^3), -15 (x^2), 71 (x), -105 (constant).

Set up synthetic division:

3 | 1   -15   71   -105
  |     3   -36   105
  ---------------------
    1   -12   35     0

So, the cubic factors as (x - 3)(x^2 -12x +35). Now, let's factor the quadratic: x^2 -12x +35.

Looking for two numbers that multiply to 35 and add to -12. Let me think: factors of 35 are 1 & 35, 5 &7. 5 and 7 add to 12. So since the middle term is -12x, both numbers should be negative: -5 and -7.

Therefore, x^2 -12x +35 factors as (x -5)(x -7).

So, putting it all together, the cubic factors as:

(x - 3)(x -5)(x -7) = 0

Therefore, the roots are x = 3, x = 5, x = 7.

Wait, but hold on. Let me check if these roots satisfy the original equation. Because sometimes when we manipulate equations, especially expanding and moving terms around, we might introduce extraneous roots or maybe lose some? Although in this case, since we just moved terms around and factored, I think they should be valid. But let me check.

First, check x=3:

Left side: (3 -3)^3 + (3 -7)^3 = 0 + (-4)^3 = -64

Right side: (2*3 -10)^3 = (6 -10)^3 = (-4)^3 = -64. So yes, equal. Good.

x=5:

Left side: (5 -3)^3 + (5 -7)^3 = (2)^3 + (-2)^3 = 8 -8 = 0

Right side: (2*5 -10)^3 = (10 -10)^3 = 0^3 = 0. Equal. Good.

x=7:

Left side: (7 -3)^3 + (7 -7)^3 = 4^3 + 0 = 64 + 0 = 64

Right side: (2*7 -10)^3 = (14 -10)^3 = 4^3 = 64. Equal. Perfect.

So all three roots are valid. Therefore, the solutions are x=3, x=5, x=7.

But wait, let me think again. The original equation was $(x - 3)^3 + (x - 7)^3 = (2x - 10)^3$. When I expanded both sides, moved everything to left, and simplified, I ended up with a cubic equation which had roots 3,5,7. But maybe there's another way to approach this problem that could have made it easier?

Alternatively, maybe there's a substitution or identity that could have been used. Let me think. For example, sometimes equations of the form a^3 + b^3 = c^3 can be approached using identities. But in this case, the left side is a sum of two cubes, and the right side is another cube. However, the standard identity for a^3 + b^3 is (a + b)(a^2 - ab + b^2), but I don't know if that helps here. Let me try.

Let me denote a = x - 3 and b = x - 7. Then the left side is a^3 + b^3. The right side is (2x -10)^3. Let me see if there's a relationship between a, b, and 2x -10.

Note that 2x -10 = 2(x -5). Also, note that a = x -3 and b = x -7. Let me compute a + b:

a + b = (x -3) + (x -7) = 2x -10. Oh! That's exactly the right side. So, the equation becomes:

a^3 + b^3 = (a + b)^3.

So, substituting back:

a^3 + b^3 = (a + b)^3.

But we know that (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3. Therefore, subtracting a^3 + b^3 from both sides:

0 = 3a^2b + 3ab^2

Which simplifies to:

0 = 3ab(a + b)

Therefore, the equation reduces to 3ab(a + b) = 0. Therefore, either a = 0, b = 0, or a + b = 0.

So, solving each case:

Case 1: a = 0 => x - 3 = 0 => x = 3

Case 2: b = 0 => x -7 = 0 => x =7

Case 3: a + b = 0 => (x -3) + (x -7) = 0 => 2x -10 = 0 => x = 5

Therefore, the solutions are x=3, x=5, x=7. Which matches the previous result.

This method seems much more straightforward. So, by recognizing that the right side is (a + b)^3 where a and b are the terms on the left, we can use the identity for the sum of cubes and simplify the equation significantly.

This shows that sometimes recognizing algebraic identities can make solving equations much easier than expanding everything out.

But since I already solved it both ways, and both methods give the same roots, I can be confident that the roots are indeed 3, 5, and 7.

Therefore, the answer is x = 3, 5, 7.

**Final Answer**
The roots are \boxed{3}, \boxed{5}, and \boxed{7}.

To solve the equation:

$$
(x - 3)^3 + (x - 7)^3 = (2x - 10)^3
$$

we begin by recognizing a key algebraic identity. Let us define:

- $ a = x - 3 $
- $ b = x - 7 $

Then the left-hand side becomes:

$$
a^3 + b^3
$$

and the right-hand side becomes:

$$
(2x - 10)^3 = (a + b)^3
$$

So the equation becomes:

$$
a^3 + b^3 = (a + b)^3
$$

We now use the identity for the cube of a sum:

$$
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
$$

Subtracting $ a^3 + b^3 $ from both sides:

$$
0 = 3a^2b + 3ab^2
$$

Factor out $ 3ab $:

$$
0 = 3ab(a + b)
$$

This gives us three possible cases:

1. $ a = 0 \Rightarrow x - 3 = 0 \Rightarrow x = 3 $
2. $ b = 0 \Rightarrow x - 7 = 0 \Rightarrow x = 7 $
3. $ a + b = 0 \Rightarrow (x - 3) + (x - 7) = 0 \Rightarrow 2x - 10 = 0 \Rightarrow x = 5 $

---

### Verification of Roots

Let’s verify each root in the original equation:

- **For $ x = 3 $:**
  $$
  (3 - 3)^3 + (3 - 7)^3 = 0 + (-4)^3 = -64
  $$
  $$
  (2 \cdot 3 - 10)^3 = (6 - 10)^3 = (-4)^3 = -64
  $$
  ✅ Equal

- **For $ x = 5 $:**
  $$
  (5 - 3)^3 + (5 - 7)^3 = 2^3 + (-2)^3 = 8 - 8 = 0
  $$
  $$
  (2 \cdot 5 - 10)^3 = (10 - 10)^3 = 0^3 = 0
  $$
  ✅ Equal

- **For $ x = 7 $:**
  $$
  (7 - 3)^3 + (7 - 7)^3 = 4^3 + 0 = 64
  $$
  $$
  (2 \cdot 7 - 10)^3 = (14 - 10)^3 = 4^3 = 64
  $$
  ✅ Equal

---

### Final Answer

The roots of the equation are:

$$
\boxed{3}, \boxed{5}, \boxed{7}
$$

Solution:
Let $a = x - 3$ and $b = x - 7.$  Then we can write the given equation as
\[a^3 + b^3 = (a + b)^3.\]Expanding, we get $a^3 + b^3 = a^3 + 3a^2 b + 3ab^2 + b^3,$ so $3a^2 b + 3ab^2 = 0,$ which factors as
\[3ab(a + b) = 0.\]Thus, $a = 0,$ $b = 0,$ or $a + b = 0.$  Then $x - 3 = 0,$ $x - 7 = 0,$ or $2x - 10 = 0.$  This gives us the roots $\boxed{3, 5, 7}.$


Explanation:
Okay, so I need to find the length of each side of a regular octagon that has the same perimeter as a regular hexagon with side length 16 cm. Let me start by recalling some basic properties of regular polygons.

First, a regular hexagon has all sides equal and all internal angles equal. Similarly, a regular octagon also has all sides equal and all internal angles equal. The perimeter of a regular polygon is just the number of sides multiplied by the length of one side.

So, let me write down what I know:

- The hexagon has side length 16 cm. Since it's regular, all sides are 16 cm. A hexagon has 6 sides, so its perimeter should be 6 times 16 cm. Let me calculate that first.

Perimeter of hexagon = 6 * 16 cm = 96 cm. Wait, 6*16... 6*10 is 60, 6*6 is 36, so 60+36=96. Yes, that's right. So the perimeter is 96 cm.

Now, the problem says that a regular octagon has the same perimeter. So the perimeter of the octagon is also 96 cm. Since an octagon has 8 sides, each side of the octagon would be the perimeter divided by 8. Let me check that.

Let me denote the side length of the octagon as 's'. Then, perimeter of octagon = 8 * s. And since this is equal to the perimeter of the hexagon, which is 96 cm, we can set up the equation:

8 * s = 96 cm

To find 's', divide both sides by 8:

s = 96 / 8 = 12 cm.

Wait, that seems straightforward. But let me make sure I didn't miss anything. The problem mentions an Asymptote figure, but since I can't see the figure, I need to rely on the description. The Asymptote code draws a hexagon with a label "16 cm" on a segment between points A and B. The code defines points A and B as dir(120) and dir(60), which are directions on the unit circle. Then M is the midpoint of A and B, and the label is placed at M, North. So, the segment AB is a side of the hexagon? Wait, in a regular hexagon, each side is equal, so if AB is a side, then its length is 16 cm. But in the Asymptote code, the coordinates are given in terms of dir(angles), which are points on the unit circle. However, in reality, the figure is drawn with those coordinates, but the actual side length is given as 16 cm. So maybe the Asymptote code is just a generic drawing, and the actual side length is 16 cm. Therefore, the perimeter calculation is correct as 6*16=96 cm. Therefore, the octagon's perimeter is also 96 cm, so each side is 96/8=12 cm. Therefore, the answer is 12 cm.

But let me think again. Is there any chance that the Asymptote figure might be misleading? For example, maybe the hexagon in the figure isn't regular? But the problem says "the regular hexagon shown here", so the figure is of a regular hexagon. The Asymptote code draws a hexagon with points dir(360)--B--A--dir(180)--dir(240)--dir(300)--cycle. Let me try to parse that.

The dir(theta) function in Asymptote gives a point on the unit circle at angle theta degrees. So dir(360) is the same as dir(0), which is (1,0). Then dir(60) is (cos(60°), sin(60°)) which is (0.5, √3/2). Similarly, dir(120) is (-0.5, √3/2). Dir(180) is (-1, 0), dir(240) is (-0.5, -√3/2), dir(300) is (0.5, -√3/2). So the figure is connecting dir(360) which is (1,0) to B which is dir(60) which is (0.5, √3/2), then to A which is dir(120) which is (-0.5, √3/2), then to dir(180) which is (-1, 0), then to dir(240) which is (-0.5, -√3/2), then to dir(300) which is (0.5, -√3/2), and back to dir(360). So connecting those points.

So this seems to be a regular hexagon? Let me check the distances between consecutive points. For example, the distance between dir(360) (which is (1,0)) and dir(60) (which is (0.5, √3/2)). The distance between (1,0) and (0.5, √3/2) is sqrt[(1 - 0.5)^2 + (0 - √3/2)^2] = sqrt[(0.5)^2 + (√3/2)^2] = sqrt[0.25 + 0.75] = sqrt[1] = 1. So in the unit circle, each side of the hexagon is length 1. But in the problem, the side length is given as 16 cm. Therefore, the Asymptote figure is just a unit circle hexagon, but scaled up so that the side length is 16 cm. Therefore, the perimeter of the hexagon is 6*16=96 cm, as I calculated before. Therefore, the octagon's perimeter is also 96 cm, so each side is 96/8=12 cm.

Therefore, the answer is 12 cm. But let me check again. Maybe there's a trick here? For example, sometimes problems mention perimeters but maybe there's something about the figure? But the problem says "a regular octagon has the same perimeter as the regular hexagon shown here with side length 16 cm." So the hexagon has side length 16 cm, so perimeter 96 cm. Therefore, octagon has perimeter 96 cm, so each side is 96/8=12 cm.

Alternatively, maybe the Asymptote figure is not a regular hexagon? Let me check the coordinates again. The Asymptote code draws dir(360)--B--A--dir(180)--dir(240)--dir(300)--cycle. Let me check if all sides are equal.

First, dir(360) to B (dir(60)): length 1 as calculated. Then B (dir(60)) to A (dir(120)): distance between dir(60) and dir(120). Let me compute that. Coordinates of dir(60) are (cos(60°), sin(60°)) = (0.5, √3/2). Coordinates of dir(120) are (cos(120°), sin(120°)) = (-0.5, √3/2). The distance between them is sqrt[(-0.5 - 0.5)^2 + (√3/2 - √3/2)^2] = sqrt[(-1)^2 + 0] = sqrt[1] = 1. Then from A (dir(120)) to dir(180): dir(180) is (-1, 0). Distance between (-0.5, √3/2) and (-1, 0): sqrt[(-1 - (-0.5))^2 + (0 - √3/2)^2] = sqrt[(-0.5)^2 + (-√3/2)^2] = sqrt[0.25 + 0.75] = sqrt[1] = 1. Then from dir(180) to dir(240): dir(240) is (-0.5, -√3/2). Distance between (-1, 0) and (-0.5, -√3/2): sqrt[(-0.5 - (-1))^2 + (-√3/2 - 0)^2] = sqrt[(0.5)^2 + (-√3/2)^2] = sqrt[0.25 + 0.75] = 1. Then from dir(240) to dir(300): dir(300) is (0.5, -√3/2). Distance between (-0.5, -√3/2) and (0.5, -√3/2): sqrt[(0.5 - (-0.5))^2 + (-√3/2 - (-√3/2))^2] = sqrt[(1)^2 + 0] = 1. Then from dir(300) back to dir(360): dir(360) is (1, 0). Distance between (0.5, -√3/2) and (1, 0): sqrt[(1 - 0.5)^2 + (0 - (-√3/2))^2] = sqrt[(0.5)^2 + (√3/2)^2] = sqrt[0.25 + 0.75] = 1. So all sides are length 1 in the unit circle. Therefore, the figure is indeed a regular hexagon. Therefore, the side length is 16 cm, so perimeter is 6*16=96 cm. Therefore, octagon's perimeter is 96 cm, so each side is 96/8=12 cm.

Therefore, the answer is 12 cm. Wait, but maybe I need to check if the Asymptote figure is drawn with some other scaling? For example, maybe the label "16 cm" is not on a side? The Asymptote code labels "16 cm" at point M, which is the midpoint of A and B. So points A and B are dir(120) and dir(60). The segment AB is a side of the hexagon? Wait, in the drawing, the hexagon is drawn as dir(360)--B--A--dir(180)--dir(240)--dir(300)--cycle. So the sides are dir(360)-B, B-A, A-dir(180), dir(180)-dir(240), dir(240)-dir(300), and dir(300)-dir(360). Therefore, AB is a side of the hexagon. Therefore, the segment AB is a side of the hexagon, and the label "16 cm" is placed at the midpoint of AB, so that segment AB is 16 cm. Therefore, the side length of the hexagon is 16 cm. Therefore, the perimeter is 6*16=96 cm. Therefore, octagon's perimeter is 96 cm, so each side is 96/8=12 cm.

Therefore, I think my answer is correct. But just to make sure, let me think again. Maybe there's a misunderstanding about the number of sides? Wait, a regular octagon has 8 sides, yes. A regular hexagon has 6 sides. So if the perimeters are equal, then 6*16 = 8*s => s= (6*16)/8 = (6/8)*16 = (3/4)*16 = 12. Yep, that's the same result.

Alternatively, maybe the problem is trying to trick me into thinking about area instead of perimeter? But the problem specifically says "perimeter". So no, that's not the case.

Alternatively, maybe the Asymptote figure is not a regular hexagon? But as we saw, all sides are equal in the Asymptote code, so it is a regular hexagon. Therefore, the side length is 16 cm.

Therefore, I think my answer is correct. The side length of the octagon is 12 cm.

**Final Answer**
The length of each side of the octagon is \boxed{12} centimeters.

To determine the side length of a regular octagon that has the same perimeter as a regular hexagon with side length 16 cm, we begin by analyzing the perimeters of both polygons.

---

### Step 1: Calculate the Perimeter of the Hexagon

A **regular hexagon** has 6 equal sides. If each side is 16 cm, then the perimeter is:

$$
\text{Perimeter of hexagon} = 6 \times 16 = 96 \text{ cm}
$$

---

### Step 2: Use the Perimeter to Find the Side Length of the Octagon

A **regular octagon** has 8 equal sides. Let the side length of the octagon be $ s $. Since the perimeter of the octagon is the same as that of the hexagon, we set up the equation:

$$
8s = 96
$$

Solving for $ s $:

$$
s = \frac{96}{8} = 12
$$

---

### Step 3: Interpret the Asymptote Diagram

The Asymptote code draws a regular hexagon with side length 16 cm, as indicated by the label "16 cm" placed at the midpoint of a side. This confirms that the hexagon in the diagram is indeed regular and has side length 16 cm. Therefore, the perimeter calculation is accurate.

---

### Final Answer

The length of each side of the regular octagon is:

$$
\boxed{12}
$$

[Assistant]
Explanation:
Okay, so I need to find the length of each side of a regular octagon that has the same perimeter as a regular hexagon with side length 16 cm. Let me start by recalling some basic properties of regular polygons.

First, a regular hexagon has all sides equal and all internal angles equal. Similarly, a regular octagon also has all sides equal and all internal angles equal. The perimeter of a regular polygon is just the number of sides multiplied by the length of one side.

So, let me write down what I know:

- The hexagon has side length 16 cm. Since it's regular, all sides are 16 cm. A hexagon has 6 sides, so its perimeter should be 6 times 16 cm. Let me calculate that first.

Perimeter of hexagon = 6 * 16 cm = 96 cm. Wait, 6*16... 6*10 is 60, 6*6 is 36, so 60+36=96. Yes, that's right. So the perimeter is 96 cm.

Now, the problem says that a regular octagon has the same perimeter. So the perimeter of the octagon is also 96 cm. Since an octagon has 8 sides, each side of the octagon would be the perimeter divided by 8. Let me check that.

Let me denote the side length of the octagon as 's'. Then, perimeter of octagon = 8 * s. And since this is equal to the perimeter of the hexagon, which is 96 cm, we can set up the equation:

8 * s = 96 cm

To find 's', divide both sides by 8:

s = 96 / 8 = 12 cm.

Wait, that seems straightforward. But let me make sure I didn't miss anything. The problem mentions an Asymptote figure, but since I can't see the figure, I need to rely on the description. The Asymptote code draws a hexagon with a label "16 cm" on a segment between points A and B. The code defines points A and B as dir(120) and dir(60), which are directions on the unit circle. Then M is the midpoint of A and B, and the label is placed at M, North. So, the segment AB is a side of the hexagon? Wait, in a regular hexagon, each side is equal, so if AB is a side, then its length is 16 cm. But in the Asymptote code, the coordinates are given in terms of dir(angles), which are points on the unit circle. However, in reality, the figure is drawn with those coordinates, but the actual side length is given as 16 cm. So maybe the Asymptote code is just a generic drawing, and the actual side length is 16 cm. Therefore, the perimeter calculation is correct as 6*16=96 cm. Therefore, the octagon's perimeter is also 96 cm, so each side is 96/8=12 cm. Therefore, the answer is 12 cm.

But let me think again. Is there any chance that the Asymptote figure might be misleading? For example, maybe the hexagon in the figure isn't regular? But the problem says "the regular hexagon shown here", so the figure is of a regular hexagon. The Asymptote code draws a hexagon with points dir(360)--B--A--dir(180)--dir(240)--dir(300)--cycle. Let me try to parse that.

The dir(theta) function in Asymptote gives a point on the unit circle at angle theta degrees. So dir(360) is the same as dir(0), which is (1,0). Then dir(60) is (cos(60°), sin(60°)) which is (0.5, √3/2). Similarly, dir(120) is (-0.5, √3/2). Dir(180) is (-1, 0), dir(240) is (-0.5, -√3/2), dir(300) is (0.5, -√3/2). So the figure is connecting dir(360) which is (1,0) to B which is dir(60) which is (0.5, √3/2), then to A which is dir(120) which is (-0.5, √3/2), then to dir(180) which is (-1, 0), then to dir(240) which is (-0.5, -√3/2), then to dir(300) which is (0.5, -√3/2), and back to dir(360). So connecting those points.

So this seems to be a regular hexagon? Let me check the distances between consecutive points. For example, the distance between dir(360) (which is (1,0)) and dir(60) (which is (0.5, √3/2)). The distance between (1,0) and (0.5, √3/2) is sqrt[(1 - 0.5)^2 + (0 - √3/2)^2] = sqrt[(0.5)^2 + (√3/2)^2] = sqrt[0.25 + 0.75] = sqrt[1] = 1. So in the unit circle, each side of the hexagon is length 1. But in the problem, the side length is given as 16 cm. Therefore, the Asymptote figure is just a unit circle hexagon, but scaled up so that the side length is 16 cm. Therefore, the perimeter of the hexagon is 6*16=96 cm, as I calculated before. Therefore, the octagon's perimeter is also 96 cm, so each side is 96/8=12 cm.

Therefore, the answer is 12 cm. But let me check again. Maybe there's a trick here? For example, sometimes problems mention perimeters but maybe there's something about the figure? But the problem says "a regular octagon has the same perimeter as the regular hexagon shown here with side length 16 cm." So the hexagon has side length 16 cm, so perimeter 96 cm. Therefore, octagon has perimeter 96 cm, so each side is 96/8=12 cm.

Alternatively, maybe the Asymptote figure is not a regular hexagon? Let me check the coordinates again. The Asymptote code draws dir(360)--B--A--dir(180)--dir(240)--dir(300)--cycle. Let me check if all sides are equal.

First, dir(360) to B (dir(60)): length 1 as calculated. Then B (dir(60)) to A (dir(120)): distance between dir(60) and dir(120). Let me compute that. Coordinates of dir(60) are (cos(60°), sin(60°)) = (0.5, √3/2). Coordinates of dir(120) are (cos(120°), sin(120°)) = (-0.5, √3/2). The distance between them is sqrt[(-0.5 - 0.5)^2 + (√3/2 - √3/2)^2] = sqrt[(-1)^2 + 0] = sqrt[1] = 1. Then from A (dir(120)) to dir(180): dir(180) is (-1, 0). Distance between (-0.5, √3/2) and (-1, 0): sqrt[(-1 - (-0.5))^2 + (0 - √3/2)^2] = sqrt[(-0.5)^2 + (-√3/2)^2] = sqrt[0.25 + 0.75] = sqrt[1] = 1. Then from dir(180) to dir(240): dir(240) is (-0.5, -√3/2). Distance between (-1, 0) and (-0.5, -√3/2): sqrt[(-0.5 - (-1))^2 + (-√3/2 - 0)^2] = sqrt[(0.5)^2 + (-√3/2)^2] = sqrt[0.25 + 0.75] = 1. Then from dir(240) to dir(300): dir(300) is (0.5, -√3/2). Distance between (-0.5, -√3/2) and (0.5, -√3/2): sqrt[(0.5 - (-0.5))^2 + (-√3/2 - (-√3/2))^2] = sqrt[(1)^2 + 0] = 1. Then from dir(300) back to dir(360): dir(360) is (1, 0). Distance between (0.5, -√3/2) and (1, 0): sqrt[(1 - 0.5)^2 + (0 - (-√3/2))^2] = sqrt[(0.5)^2 + (√3/2)^2] = sqrt[0.25 + 0.75] = 1. So all sides are length 1 in the unit circle. Therefore, the figure is indeed a regular hexagon. Therefore, the side length is 16 cm, so perimeter is 6*16=96 cm. Therefore, octagon's perimeter is 96 cm, so each side is 96/8=12 cm.

Therefore, the answer is 12 cm. Wait, but maybe I need to check if the Asymptote figure is drawn with some other scaling? For example, maybe the label "16 cm" is not on a side? The Asymptote code labels "16 cm" at point M, which is the midpoint of A and B. So points A and B are dir(120) and dir(60). The segment AB is a side of the hexagon? Wait, in the drawing, the hexagon is drawn as dir(360)--B--A--dir(180)--dir(240)--dir(300)--cycle. So the sides are dir(360)-B, B-A, A-dir(180), dir(180)-dir(240), dir(240)-dir(300), and dir(300)-dir(360). Therefore, AB is a side of the hexagon. Therefore, the segment AB is a side of the hexagon, and the label "16 cm" is placed at the midpoint of AB, so that segment AB is 16 cm. Therefore, the side length of the hexagon is 16 cm. Therefore, the perimeter is 6*16=96 cm. Therefore, octagon's perimeter is 96 cm, so each side is 96/8=12 cm.

Therefore, I think my answer is correct. But just to make sure, let me think again. Maybe there's a misunderstanding about the number of sides? Wait, a regular octagon has 8 sides, yes. A regular hexagon has 6 sides. So if the perimeters are equal, then 6*16 = 8*s => s= (6*16)/8 = (6/8)*16 = (3/4)*16 = 12. Yep, that's the same result.

Alternatively, maybe the problem is trying to trick me into thinking about area instead of perimeter? But the problem specifically says "perimeter". So no, that's not the case.

Alternatively, maybe the Asymptote figure is not a regular hexagon? But as we saw, all sides are equal in the Asymptote code, so it is a regular hexagon. Therefore, the side length is 16 cm.

Therefore, I think my answer is correct. The side length of the octagon is 12 cm.

**Final Answer**
The length of each side of the octagon is \boxed{12} centimeters.

To determine the side length of a regular octagon that has the same perimeter as a regular hexagon with side length 16 cm, we begin by analyzing the perimeters of both polygons.

---

### Step 1: Calculate the Perimeter of the Hexagon

A **regular hexagon** has 6 equal sides. If each side is 16 cm, then the perimeter is:

$$
\text{Perimeter of hexagon} = 6 \times 16 = 96 \text{ cm}
$$

---

### Step 2: Use the Perimeter to Find the Side Length of the Octagon

A **regular octagon** has 8 equal sides. Let the side length of the octagon be $ s $. Since the perimeter of the octagon is the same as that of the hexagon, we set up the equation:

$$
8s = 96
$$

Solving for $ s $:

$$
s = \frac{96}{8} = 12
$$

---

### Step 3: Interpret the Asymptote Diagram

The Asymptote code draws a regular hexagon with side length 16 cm, as indicated by the label "16 cm" placed at the midpoint of a side. This confirms that the hexagon in the diagram is indeed regular and has side length 16 cm. Therefore, the perimeter calculation is accurate.

---

### Final Answer

The length of each side of the regular octagon is:

$$
\boxed{12}
$$

Solution: The hexagon has a side length of 16 centimeters, so its perimeter is $16\times 6 = 96$ centimeters. Since the octagon and the hexagon have the same perimeter, it follows that each side of the octagon has a length of $96/8 = \boxed{12}$ centimeters.


Explanation:
Okay, so I need to solve this inequality: -4 < 2(x - 1) < 8. And then express the solution in the form a < x < b, and find a + b. Alright, let me think about how to approach this.

First, I remember that when dealing with compound inequalities like this, I can solve them step by step by isolating x. The inequality is already set up with 2(x - 1) in the middle, so maybe I can start by dividing all parts of the inequality by 2 to simplify it. Let me try that.

So, dividing each part by 2:

-4 divided by 2 is -2, right? Then the middle part is 2(x - 1) divided by 2, which simplifies to just (x - 1). And then 8 divided by 2 is 4. So now the inequality becomes:

-2 < x - 1 < 4.

Okay, that seems simpler. Now I need to solve for x. To do that, I should get rid of the -1 that's being subtracted from x. How do I do that? Well, I can add 1 to all three parts of the inequality. Let me write that down.

Adding 1 to each part:

-2 + 1 < x - 1 + 1 < 4 + 1.

Calculating each part:

-2 + 1 is -1, then x - 1 + 1 simplifies to x, and 4 + 1 is 5. So now the inequality is:

-1 < x < 5.

Wait, so that means x is between -1 and 5? Let me check if I did that correctly. Let me go back through the steps.

Original inequality: -4 < 2(x - 1) < 8.

Divide all parts by 2: -2 < x - 1 < 4. That seems right because dividing each part by 2 maintains the inequality direction since 2 is positive. Then adding 1 to all parts: -2 + 1 is -1, and 4 + 1 is 5. So yes, that gives -1 < x < 5. Therefore, a is -1 and b is 5. Then a + b would be -1 + 5, which is 4. Is that correct?

Wait, let me verify by plugging in some numbers. Let me check if x = 0, which is between -1 and 5. Plugging into the original inequality: 2(0 - 1) = 2(-1) = -2. So the original inequality is -4 < -2 < 8. That's true because -4 is less than -2 and -2 is less than 8. Okay, that works.

What about x = -1? Let me check the boundary. If x = -1, then 2(-1 - 1) = 2(-2) = -4. The original inequality is -4 < -4 < 8. But -4 is not greater than -4, so x = -1 is not included. Similarly, if x = 5, then 2(5 - 1) = 2(4) = 8. The original inequality is -4 < 8 < 8? No, because 8 is not less than 8. So x = 5 is also not included. Therefore, the solution is indeed -1 < x < 5. So a is -1 and b is 5, so a + b is 4. That seems right.

But let me try another number outside the interval to make sure. Let's take x = -2. Then 2(-2 - 1) = 2(-3) = -6. The original inequality is -4 < -6 < 8? No, because -6 is less than -4, so that's not true. So x = -2 is not a solution, which is consistent with our solution.

How about x = 6? Then 2(6 - 1) = 2(5) = 10. The original inequality is -4 < 10 < 8? No, because 10 is not less than 8. So that's not a solution either. So our interval seems correct.

Wait, but let me think again. When I divided by 2, I had to make sure that 2 is positive, which it is, so the direction of the inequalities doesn't change. Then adding 1 to all parts is straightforward. So I think the steps are correct.

Alternatively, maybe I can solve the inequality without dividing first. Let me try that approach to cross-verify.

Original inequality: -4 < 2(x - 1) < 8.

First, maybe I can subtract 2 from all parts? Wait, no, the middle term is 2(x - 1). Alternatively, maybe I can split the compound inequality into two separate inequalities:

-4 < 2(x - 1) and 2(x - 1) < 8.

Then solve each one separately.

Starting with the first inequality: -4 < 2(x - 1).

Divide both sides by 2: -2 < x - 1.

Then add 1 to both sides: -2 + 1 < x => -1 < x.

Now the second inequality: 2(x - 1) < 8.

Divide both sides by 2: x - 1 < 4.

Add 1 to both sides: x < 5.

So combining both inequalities: -1 < x < 5. Same result as before. So that confirms that the solution is indeed -1 < x < 5. Therefore, a is -1 and b is 5, so a + b is 4.

Therefore, the answer should be 4. Wait, but let me check once more. Maybe I made a mistake in arithmetic?

Original problem: -4 < 2(x - 1) < 8.

Divide by 2: -2 < x - 1 < 4. Then add 1: -1 < x < 5. Yes. So a is -1, b is 5. Sum is 4. Seems straightforward.

Alternatively, maybe I can think of it as solving for x in the middle. Let me try another way. Let me expand the original inequality.

Original inequality: -4 < 2(x - 1) < 8.

First, distribute the 2: -4 < 2x - 2 < 8.

Then, add 2 to all parts: -4 + 2 < 2x - 2 + 2 < 8 + 2.

Which simplifies to: -2 < 2x < 10.

Then divide all parts by 2: -1 < x < 5. Same result again. So that's three different methods leading to the same answer. So I think that's solid.

Therefore, the value of a + b is -1 + 5 = 4. So the answer is 4. I think that's correct. I don't see any mistakes in my reasoning. All the steps check out, and multiple methods lead to the same conclusion. So I feel confident that the answer is 4.

**Final Answer**
The value of $a + b$ is \boxed{4}.

To solve the inequality:

$$
-4 < 2(x - 1) < 8
$$

we aim to isolate $ x $ and express the solution in the form $ a < x < b $, then compute $ a + b $.

---

### Step 1: Simplify the inequality

We begin by dividing all parts of the inequality by 2, since 2 is a positive number and does not change the direction of the inequality:

$$
\frac{-4}{2} < \frac{2(x - 1)}{2} < \frac{8}{2}
$$

$$
-2 < x - 1 < 4
$$

---

### Step 2: Solve for $ x $

Now, we isolate $ x $ by adding 1 to all parts of the inequality:

$$
-2 + 1 < x - 1 + 1 < 4 + 1
$$

$$
-1 < x < 5
$$

---

### Step 3: Identify $ a $ and $ b $

From the simplified inequality $ -1 < x < 5 $, we identify:

- $ a = -1 $
- $ b = 5 $

---

### Step 4: Compute $ a + b $

$$
a + b = -1 + 5 = 4
$$

---

### Final Answer

$$
\boxed{4}
$$

[Assistant]
Explanation:
Okay, so I need to solve this inequality: -4 < 2(x - 1) < 8. And then express the solution in the form a < x < b, and find a + b. Alright, let me think about how to approach this.

First, I remember that when dealing with compound inequalities like this, I can solve them step by step by isolating x. The inequality is already set up with 2(x - 1) in the middle, so maybe I can start by dividing all parts of the inequality by 2 to simplify it. Let me try that.

So, dividing each part by 2:

-4 divided by 2 is -2, right? Then the middle part is 2(x - 1) divided by 2, which simplifies to just (x - 1). And then 8 divided by 2 is 4. So now the inequality becomes:

-2 < x - 1 < 4.

Okay, that seems simpler. Now I need to solve for x. To do that, I should get rid of the -1 that's being subtracted from x. How do I do that? Well, I can add 1 to all three parts of the inequality. Let me write that down.

Adding 1 to each part:

-2 + 1 < x - 1 + 1 < 4 + 1.

Calculating each part:

-2 + 1 is -1, then x - 1 + 1 simplifies to x, and 4 + 1 is 5. So now the inequality is:

-1 < x < 5.

Wait, so that means x is between -1 and 5? Let me check if I did that correctly. Let me go back through the steps.

Original inequality: -4 < 2(x - 1) < 8.

Divide all parts by 2: -2 < x - 1 < 4. That seems right because dividing each part by 2 maintains the inequality direction since 2 is positive. Then adding 1 to all parts: -2 + 1 is -1, and 4 + 1 is 5. So yes, that gives -1 < x < 5. Therefore, a is -1 and b is 5. Then a + b would be -1 + 5, which is 4. Is that correct?

Wait, let me verify by plugging in some numbers. Let me check if x = 0, which is between -1 and 5. Plugging into the original inequality: 2(0 - 1) = 2(-1) = -2. So the original inequality is -4 < -2 < 8. That's true because -4 is less than -2 and -2 is less than 8. Okay, that works.

What about x = -1? Let me check the boundary. If x = -1, then 2(-1 - 1) = 2(-2) = -4. The original inequality is -4 < -4 < 8. But -4 is not greater than -4, so x = -1 is not included. Similarly, if x = 5, then 2(5 - 1) = 2(4) = 8. The original inequality is -4 < 8 < 8? No, because 8 is not less than 8. So x = 5 is also not included. Therefore, the solution is indeed -1 < x < 5. So a is -1 and b is 5, so a + b is 4. That seems right.

But let me try another number outside the interval to make sure. Let's take x = -2. Then 2(-2 - 1) = 2(-3) = -6. The original inequality is -4 < -6 < 8? No, because -6 is less than -4, so that's not true. So x = -2 is not a solution, which is consistent with our solution.

How about x = 6? Then 2(6 - 1) = 2(5) = 10. The original inequality is -4 < 10 < 8? No, because 10 is not less than 8. So that's not a solution either. So our interval seems correct.

Wait, but let me think again. When I divided by 2, I had to make sure that 2 is positive, which it is, so the direction of the inequalities doesn't change. Then adding 1 to all parts is straightforward. So I think the steps are correct.

Alternatively, maybe I can solve the inequality without dividing first. Let me try that approach to cross-verify.

Original inequality: -4 < 2(x - 1) < 8.

First, maybe I can subtract 2 from all parts? Wait, no, the middle term is 2(x - 1). Alternatively, maybe I can split the compound inequality into two separate inequalities:

-4 < 2(x - 1) and 2(x - 1) < 8.

Then solve each one separately.

Starting with the first inequality: -4 < 2(x - 1).

Divide both sides by 2: -2 < x - 1.

Then add 1 to both sides: -2 + 1 < x => -1 < x.

Now the second inequality: 2(x - 1) < 8.

Divide both sides by 2: x - 1 < 4.

Add 1 to both sides: x < 5.

So combining both inequalities: -1 < x < 5. Same result as before. So that confirms that the solution is indeed -1 < x < 5. Therefore, a is -1 and b is 5, so a + b is 4.

Therefore, the answer should be 4. Wait, but let me check once more. Maybe I made a mistake in arithmetic?

Original problem: -4 < 2(x - 1) < 8.

Divide by 2: -2 < x - 1 < 4. Then add 1: -1 < x < 5. Yes. So a is -1, b is 5. Sum is 4. Seems straightforward.

Alternatively, maybe I can think of it as solving for x in the middle. Let me try another way. Let me expand the original inequality.

Original inequality: -4 < 2(x - 1) < 8.

First, distribute the 2: -4 < 2x - 2 < 8.

Then, add 2 to all parts: -4 + 2 < 2x - 2 + 2 < 8 + 2.

Which simplifies to: -2 < 2x < 10.

Then divide all parts by 2: -1 < x < 5. Same result again. So that's three different methods leading to the same answer. So I think that's solid.

Therefore, the value of a + b is -1 + 5 = 4. So the answer is 4. I think that's correct. I don't see any mistakes in my reasoning. All the steps check out, and multiple methods lead to the same conclusion. So I feel confident that the answer is 4.

**Final Answer**
The value of $a + b$ is \boxed{4}.

To solve the inequality:

$$
-4 < 2(x - 1) < 8
$$

we aim to isolate $ x $ and express the solution in the form $ a < x < b $, then compute $ a + b $.

---

### Step 1: Simplify the inequality

We begin by dividing all parts of the inequality by 2, since 2 is a positive number and does not change the direction of the inequality:

$$
\frac{-4}{2} < \frac{2(x - 1)}{2} < \frac{8}{2}
$$

$$
-2 < x - 1 < 4
$$

---

### Step 2: Solve for $ x $

Now, we isolate $ x $ by adding 1 to all parts of the inequality:

$$
-2 + 1 < x - 1 + 1 < 4 + 1
$$

$$
-1 < x < 5
$$

---

### Step 3: Identify $ a $ and $ b $

From the simplified inequality $ -1 < x < 5 $, we identify:

- $ a = -1 $
- $ b = 5 $

---

### Step 4: Compute $ a + b $

$$
a + b = -1 + 5 = 4
$$

---

### Final Answer

$$
\boxed{4}
$$

Solution: Since everything in sight is even, we should begin by dividing by 2.  That gives \[-2<x-1<4.\] To isolate $x$, we add 1, so \[-1<x<5.\] Since $a=-1$ and $b=5$, we get $a+b=-1+5=\boxed{4}$.