Explanation:
Okay, so I need to find p(8) where p(x) is a degree 5 polynomial that satisfies p(n) = n/(n² - 1) for n = 2, 3, 4, 5, 6, 7. Hmm, let me think about how to approach this.

First, I remember that if a polynomial of degree d passes through d+1 points, then it's uniquely determined. But here, we have a degree 5 polynomial passing through 6 points (from n=2 to n=7). So that makes sense, since 5+1=6. So p(x) is uniquely determined by these conditions. But how do I find p(8)?

One method I recall is using Lagrange interpolation, but with 6 points, that might get complicated. Maybe there's a smarter way. Let me think.

Alternatively, since the polynomial p(x) agrees with the function f(x) = x/(x² - 1) at x = 2, 3, 4, 5, 6, 7, maybe I can consider the difference between p(x) and f(x). But wait, f(x) isn't a polynomial, it's a rational function. However, maybe I can construct a polynomial that relates them?

Let me think. Let me define a new polynomial q(x) = (x² - 1)p(x) - x. Then, for x = 2, 3, 4, 5, 6, 7, we have:

q(n) = (n² - 1)p(n) - n = (n² - 1)*(n/(n² - 1)) - n = n - n = 0.

So q(x) has roots at x = 2, 3, 4, 5, 6, 7. Therefore, q(x) is a polynomial that has these six roots. But what's the degree of q(x)?

Well, p(x) is degree 5, so (x² - 1)p(x) is degree 5 + 2 = 7. Then subtracting x, which is degree 1, so q(x) is degree 7. Therefore, q(x) is a degree 7 polynomial with roots at 2, 3, 4, 5, 6, 7. But since it's degree 7, there must be another root. Let me denote that root as r. So we can write:

q(x) = (x - 2)(x - 3)(x - 4)(x - 5)(x - 6)(x - 7)(x - r)

But we need to find r. How?

Well, since q(x) = (x² - 1)p(x) - x, and p(x) is a degree 5 polynomial, then q(x) is degree 7 as we said. But maybe we can find r by considering the leading coefficients?

Let me compute the leading coefficient of q(x). Let's see:

The leading term of (x² - 1)p(x) is x² * (leading term of p(x)). Since p(x) is degree 5, let's say p(x) = a x^5 + ... Then (x² - 1)p(x) would have leading term a x^7. Then subtracting x, which doesn't affect the leading term. Therefore, the leading term of q(x) is a x^7.

On the other hand, the leading term of (x - 2)(x - 3)...(x - 7)(x - r) is x^7 multiplied by the product of the leading coefficients of each factor, which is 1*1*...*1 = 1. Therefore, the leading coefficient of q(x) is 1*(product of the roots' coefficients?), Wait, no. Wait, the leading coefficient of a polynomial (x - a1)(x - a2)...(x - an) is 1. So the leading term is x^n. Therefore, the leading coefficient of q(x) is 1. Therefore, comparing to the leading term of q(x) from the other expression, which is a x^7. Therefore, we have a = 1? Wait, but p(x) is degree 5, so a is the leading coefficient of p(x). But how do we know a?

Wait, maybe we need another condition. Wait, but maybe we can find r by considering another value? Wait, but we don't have any other values of p(x). However, maybe we can use the fact that q(x) is equal to (x² - 1)p(x) - x, so if we can evaluate q(x) at some other point, maybe we can find r?

But we don't have any other information about p(x). Wait, but maybe we can use the fact that q(x) is a degree 7 polynomial with roots at 2 through 7 and another root r. Therefore, we can write:

q(x) = (x - 2)(x - 3)(x - 4)(x - 5)(x - 6)(x - 7)(x - r)

But also, q(x) = (x² - 1)p(x) - x. Therefore, if we can find another value of q(x), we can solve for r. But how?

Wait, maybe we can plug in x = 1? Let me try that. Let me compute q(1):

q(1) = (1² - 1)p(1) - 1 = (0)*p(1) - 1 = -1.

On the other hand, q(1) is also equal to (1 - 2)(1 - 3)(1 - 4)(1 - 5)(1 - 6)(1 - 7)(1 - r). Let me compute that:

(1 - 2) = -1

(1 - 3) = -2

(1 - 4) = -3

(1 - 5) = -4

(1 - 6) = -5

(1 - 7) = -6

(1 - r) = (1 - r)

So multiplying all together:

(-1)*(-2)*(-3)*(-4)*(-5)*(-6)*(1 - r)

Let me compute the product of the first six terms:

(-1)*(-2) = 2

2*(-3) = -6

-6*(-4) = 24

24*(-5) = -120

-120*(-6) = 720

So the product is 720*(1 - r)

Therefore, q(1) = 720*(1 - r) = -1

So solving for r:

720*(1 - r) = -1 => 1 - r = -1/720 => r = 1 + 1/720 = 721/720

Therefore, r = 721/720.

So now we have q(x) = (x - 2)(x - 3)(x - 4)(x - 5)(x - 6)(x - 7)(x - 721/720)

Therefore, we can write:

(x² - 1)p(x) - x = (x - 2)(x - 3)(x - 4)(x - 5)(x - 6)(x - 7)(x - 721/720)

Therefore, solving for p(x):

p(x) = [ (x - 2)(x - 3)(x - 4)(x - 5)(x - 6)(x - 7)(x - 721/720) + x ] / (x² - 1)

Therefore, to find p(8), we can plug x = 8 into this expression.

So let's compute p(8):

First, compute the numerator:

N = (8 - 2)(8 - 3)(8 - 4)(8 - 5)(8 - 6)(8 - 7)(8 - 721/720) + 8

Let me compute each term:

(8 - 2) = 6

(8 - 3) = 5

(8 - 4) = 4

(8 - 5) = 3

(8 - 6) = 2

(8 - 7) = 1

(8 - 721/720) = 8 - 721/720. Let me convert 8 to 5760/720, so 5760/720 - 721/720 = (5760 - 721)/720 = 5039/720

Therefore, the product of the first six terms is 6*5*4*3*2*1 = 720

Then multiply by 5039/720:

720 * (5039/720) = 5039

Then add 8: N = 5039 + 8 = 5047

Then the denominator is (8² - 1) = 64 - 1 = 63

Therefore, p(8) = 5047 / 63

Wait, let me check that again. Wait, the numerator is [product + 8], which is 5039 + 8 = 5047? Wait, no. Wait, the product of the first six terms is 6*5*4*3*2*1 = 720, then multiplied by (8 - 721/720) which is 5039/720, so 720*(5039/720) = 5039. Then add x, which is 8? Wait, no. Wait, the original expression is:

N = [ (product) + x ] ?

Wait, no. Wait, the original expression is:

q(x) = (x² - 1)p(x) - x = product*(x - r)

Therefore, solving for p(x):

(x² - 1)p(x) = q(x) + x = product*(x - r) + x

Therefore, p(x) = [ product*(x - r) + x ] / (x² - 1)

Therefore, when x = 8, p(8) = [ product*(8 - r) + 8 ] / (8² - 1)

Wait, but earlier I said that product*(8 - r) is 720*(5039/720) = 5039, then adding 8? Wait, no. Wait, the product is (x - 2)...(x - 7) which at x=8 is 6*5*4*3*2*1 = 720, then multiplied by (x - r) which is (8 - r) = 5039/720. Therefore, the entire product term is 720*(5039/720) = 5039. Then adding x, which is 8? Wait, no. Wait, the formula is:

q(x) = product*(x - r) = (x² - 1)p(x) - x

Therefore, (x² - 1)p(x) = q(x) + x = product*(x - r) + x

Therefore, p(x) = [ product*(x - r) + x ] / (x² - 1)

Therefore, when x = 8, p(8) = [ product*(8 - r) + 8 ] / (8² - 1)

But product*(8 - r) is 720*(5039/720) = 5039, then adding 8 gives 5039 + 8 = 5047. Then divide by 63. So p(8) = 5047 / 63.

Wait, but let me check if that reduces. Let me compute 5047 divided by 63.

First, 63*80 = 5040. Therefore, 5047 - 5040 = 7. Therefore, 5047 / 63 = 80 + 7/63 = 80 + 1/9 = 80.111...

But the question says p(x) is a polynomial, so p(8) should be a rational number. But maybe 5047/63 can be simplified? Let me check if 5047 and 63 have common factors.

63 factors into 7*9. Let me check if 7 divides 5047. 7*721 = 5047? Wait, 7*700 = 4900, 7*21 = 147, so 4900 + 147 = 5047. Yes! So 5047 = 7*721. Therefore, 5047 / 63 = (7*721)/(7*9) ) = 721/9. Therefore, p(8) = 721/9.

Wait, that's better. So 721 divided by 9. Let me check 9*80 = 720, so 721/9 = 80 + 1/9, which is 80.111..., which matches the earlier decimal. So 721/9 is the simplified fraction.

But let me check my steps again to make sure I didn't make a mistake.

First, we had q(x) = (x² - 1)p(x) - x, which has roots at 2 through 7, so we wrote it as (x - 2)...(x - 7)(x - r). Then evaluated at x=1, found that q(1) = -1, which led us to find r = 721/720. Then, when computing p(8), we calculated the product (8 - 2)...(8 - 7) = 6! = 720, multiplied by (8 - r) = 8 - 721/720 = (5760 - 721)/720 = 5039/720. Then 720*(5039/720) = 5039. Then adding x, which is 8? Wait, no. Wait, the formula is [product*(x - r) + x] / (x² - 1). Wait, so product*(x - r) is 5039, then adding x which is 8? Wait, no, the formula is:

(x² - 1)p(x) = product*(x - r) + x. Therefore, p(x) = [product*(x - r) + x]/(x² - 1). Therefore, when x = 8, it's [product*(8 - r) + 8]/(8² - 1). So product*(8 - r) is 5039, then adding 8 gives 5039 + 8 = 5047. Then divide by 63. Then 5047 / 63 = 721/9. So that seems correct.

But let me check if 721/9 is the correct answer. Let me check if there's another way to verify.

Alternatively, maybe I can check with another value. For example, let me check if p(1) can be calculated? Wait, but we don't have p(1) given. However, since we have q(x) = (x² - 1)p(x) - x, then p(1) would be [q(1) + 1]/(1² - 1). But denominator is zero, so that's undefined. Which makes sense because p(x) is a polynomial, but when we plug x=1 into the original equation, we have division by zero. So that's okay.

Alternatively, maybe check for another value. Let me try x=0. Wait, but we don't have p(0) given. Maybe not helpful.

Alternatively, maybe check if the answer 721/9 makes sense. Let me think. Let me compute 721 divided by 9. 9*80 = 720, so 721/9 = 80 + 1/9 ≈ 80.111...

Alternatively, maybe there's a different approach. Let me think.

Another approach: Since p(x) is a degree 5 polynomial, and we know its values at 6 points, maybe we can use finite differences or something else? But that might be more complicated.

Alternatively, consider that the function f(x) = x/(x² - 1) can be written as f(x) = 1/(x - 1) - 1/(x + 1). Let me check:

1/(x - 1) - 1/(x + 1) = [ (x + 1) - (x - 1) ] / (x² - 1) ) = [2]/(x² - 1). Wait, that's 2/(x² - 1). So f(x) = x/(x² - 1) is not directly that. But maybe there's a relation.

Alternatively, maybe consider that f(x) = [x/(x² - 1)] = [1/(x - 1)] - [1/(x + 1)] / 2? Wait, let me check:

Let me compute [1/(x - 1) - 1/(x + 1)] / 2 = [ (x + 1 - x + 1) / (x² - 1) ] / 2 = [2 / (x² - 1)] / 2 = 1/(x² - 1). So that's different from f(x). So f(x) is x/(x² - 1). Maybe not helpful directly.

Alternatively, think about constructing a polynomial that interpolates these points. But with Lagrange interpolation, but with 6 points, it's going to be tedious. However, maybe there's a pattern.

Alternatively, think about the fact that p(x) - f(x) has roots at 2 through 7, but f(x) isn't a polynomial. However, we already did that approach earlier, which led us to the answer. So maybe that's the correct way.

Alternatively, maybe check if 721/9 is correct by plugging into the equation. Let me check if with p(8) = 721/9, then (8² - 1)p(8) - 8 should equal q(8) = (8 - 2)...(8 - 7)(8 - r). Let me compute both sides.

Left side: (64 - 1)*(721/9) - 8 = 63*(721/9) - 8 = 7*721 - 8 = 5047 - 8 = 5039.

Right side: product*(8 - r) = 720*(8 - 721/720) = 720*( (5760 - 721)/720 ) = 5039. Which matches. Therefore, that checks out. Therefore, p(8) = 721/9 is correct.

Therefore, the answer is 721/9. Let me check if that reduces further. 721 divided by... Let me check if 721 is divisible by any number. 721 ÷ 7 = 103, since 7*100=700, 7*3=21, so 7*103=721. Therefore, 721 = 7*103. Then 9 is 3². So no common factors. Therefore, 721/9 is reduced.

Therefore, the answer is 721/9. So boxed, it's \boxed{\dfrac{721}{9}}.

**Final Answer**
\boxed{\dfrac{721}{9}}

To solve the problem, we are given a polynomial $ p(x) $ of degree 5 such that:

$$
p(n) = \frac{n}{n^2 - 1} \quad \text{for } n = 2, 3, 4, 5, 6, 7
$$

We are asked to find $ p(8) $.

---

### Step 1: Define a Related Polynomial

Let us define a new polynomial:

$$
q(x) = (x^2 - 1)p(x) - x
$$

This is a polynomial of degree 7, since $ p(x) $ is degree 5 and $ x^2 - 1 $ is degree 2. The subtraction of $ x $ does not change the degree.

Now, observe that for $ x = 2, 3, 4, 5, 6, 7 $, we have:

$$
q(n) = (n^2 - 1)p(n) - n = (n^2 - 1)\cdot\frac{n}{n^2 - 1} - n = n - n = 0
$$

So, $ q(x) $ has roots at $ x = 2, 3, 4, 5, 6, 7 $. Therefore, we can write:

$$
q(x) = (x - 2)(x - 3)(x - 4)(x - 5)(x - 6)(x - 7)(x - r)
$$

for some unknown root $ r $.

---

### Step 2: Determine the Missing Root

We can find $ r $ by evaluating $ q(x) $ at a point not in the given set. A natural choice is $ x = 1 $, since $ x^2 - 1 = 0 $ at $ x = 1 $, and we can compute $ q(1) $ in two ways.

From the definition:

$$
q(1) = (1^2 - 1)p(1) - 1 = 0 \cdot p(1) - 1 = -1
$$

From the factored form:

$$
q(1) = (1 - 2)(1 - 3)(1 - 4)(1 - 5)(1 - 6)(1 - 7)(1 - r)
$$

Compute the product:

$$
(-1)(-2)(-3)(-4)(-5)(-6)(1 - r) = 720(1 - r)
$$

So:

$$
720(1 - r) = -1 \Rightarrow 1 - r = -\frac{1}{720} \Rightarrow r = 1 + \frac{1}{720} = \frac{721}{720}
$$

---

### Step 3: Use the Polynomial to Find $ p(8) $

We now have:

$$
q(x) = (x - 2)(x - 3)(x - 4)(x - 5)(x - 6)(x - 7)\left(x - \frac{721}{720}\right)
$$

Recall that:

$$
q(x) = (x^2 - 1)p(x) - x \Rightarrow p(x) = \frac{q(x) + x}{x^2 - 1}
$$

So, to find $ p(8) $, compute:

$$
p(8) = \frac{q(8) + 8}{8^2 - 1} = \frac{q(8) + 8}{63}
$$

Now compute $ q(8) $:

$$
q(8) = (8 - 2)(8 - 3)(8 - 4)(8 - 5)(8 - 6)(8 - 7)\left(8 - \frac{721}{720}\right)
$$

Compute each term:

- $ 8 - 2 = 6 $
- $ 8 - 3 = 5 $
- $ 8 - 4 = 4 $
- $ 8 - 5 = 3 $
- $ 8 - 6 = 2 $
- $ 8 - 7 = 1 $
- $ 8 - \frac{721}{720} = \frac{5760 - 721}{720} = \frac{5039}{720} $

So:

$$
q(8) = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 \cdot \frac{5039}{720} = 720 \cdot \frac{5039}{720} = 5039
$$

Then:

$$
p(8) = \frac{5039 + 8}{63} = \frac{5047}{63}
$$

Now simplify:

$$
\frac{5047}{63} = \frac{7 \cdot 721}{7 \cdot 9} = \frac{721}{9}
$$

---

### Final Answer

$$
\boxed{\dfrac{721}{9}}
$$

[Assistant]
Explanation:
Okay, so I need to find p(8) where p(x) is a degree 5 polynomial that satisfies p(n) = n/(n² - 1) for n = 2, 3, 4, 5, 6, 7. Hmm, let me think about how to approach this.

First, I remember that if a polynomial of degree d passes through d+1 points, then it's uniquely determined. But here, we have a degree 5 polynomial passing through 6 points (from n=2 to n=7). So that makes sense, since 5+1=6. So p(x) is uniquely determined by these conditions. But how do I find p(8)?

One method I recall is using Lagrange interpolation, but with 6 points, that might get complicated. Maybe there's a smarter way. Let me think.

Alternatively, since the polynomial p(x) agrees with the function f(x) = x/(x² - 1) at x = 2, 3, 4, 5, 6, 7, maybe I can consider the difference between p(x) and f(x). But wait, f(x) isn't a polynomial, it's a rational function. However, maybe I can construct a polynomial that relates them?

Let me think. Let me define a new polynomial q(x) = (x² - 1)p(x) - x. Then, for x = 2, 3, 4, 5, 6, 7, we have:

q(n) = (n² - 1)p(n) - n = (n² - 1)*(n/(n² - 1)) - n = n - n = 0.

So q(x) has roots at x = 2, 3, 4, 5, 6, 7. Therefore, q(x) is a polynomial that has these six roots. But what's the degree of q(x)?

Well, p(x) is degree 5, so (x² - 1)p(x) is degree 5 + 2 = 7. Then subtracting x, which is degree 1, so q(x) is degree 7. Therefore, q(x) is a degree 7 polynomial with roots at 2, 3, 4, 5, 6, 7. But since it's degree 7, there must be another root. Let me denote that root as r. So we can write:

q(x) = (x - 2)(x - 3)(x - 4)(x - 5)(x - 6)(x - 7)(x - r)

But we need to find r. How?

Well, since q(x) = (x² - 1)p(x) - x, and p(x) is a degree 5 polynomial, then q(x) is degree 7 as we said. But maybe we can find r by considering the leading coefficients?

Let me compute the leading coefficient of q(x). Let's see:

The leading term of (x² - 1)p(x) is x² * (leading term of p(x)). Since p(x) is degree 5, let's say p(x) = a x^5 + ... Then (x² - 1)p(x) would have leading term a x^7. Then subtracting x, which doesn't affect the leading term. Therefore, the leading term of q(x) is a x^7.

On the other hand, the leading term of (x - 2)(x - 3)...(x - 7)(x - r) is x^7 multiplied by the product of the leading coefficients of each factor, which is 1*1*...*1 = 1. Therefore, the leading coefficient of q(x) is 1*(product of the roots' coefficients?), Wait, no. Wait, the leading coefficient of a polynomial (x - a1)(x - a2)...(x - an) is 1. So the leading term is x^n. Therefore, the leading coefficient of q(x) is 1. Therefore, comparing to the leading term of q(x) from the other expression, which is a x^7. Therefore, we have a = 1? Wait, but p(x) is degree 5, so a is the leading coefficient of p(x). But how do we know a?

Wait, maybe we need another condition. Wait, but maybe we can find r by considering another value? Wait, but we don't have any other values of p(x). However, maybe we can use the fact that q(x) is equal to (x² - 1)p(x) - x, so if we can evaluate q(x) at some other point, maybe we can find r?

But we don't have any other information about p(x). Wait, but maybe we can use the fact that q(x) is a degree 7 polynomial with roots at 2 through 7 and another root r. Therefore, we can write:

q(x) = (x - 2)(x - 3)(x - 4)(x - 5)(x - 6)(x - 7)(x - r)

But also, q(x) = (x² - 1)p(x) - x. Therefore, if we can find another value of q(x), we can solve for r. But how?

Wait, maybe we can plug in x = 1? Let me try that. Let me compute q(1):

q(1) = (1² - 1)p(1) - 1 = (0)*p(1) - 1 = -1.

On the other hand, q(1) is also equal to (1 - 2)(1 - 3)(1 - 4)(1 - 5)(1 - 6)(1 - 7)(1 - r). Let me compute that:

(1 - 2) = -1

(1 - 3) = -2

(1 - 4) = -3

(1 - 5) = -4

(1 - 6) = -5

(1 - 7) = -6

(1 - r) = (1 - r)

So multiplying all together:

(-1)*(-2)*(-3)*(-4)*(-5)*(-6)*(1 - r)

Let me compute the product of the first six terms:

(-1)*(-2) = 2

2*(-3) = -6

-6*(-4) = 24

24*(-5) = -120

-120*(-6) = 720

So the product is 720*(1 - r)

Therefore, q(1) = 720*(1 - r) = -1

So solving for r:

720*(1 - r) = -1 => 1 - r = -1/720 => r = 1 + 1/720 = 721/720

Therefore, r = 721/720.

So now we have q(x) = (x - 2)(x - 3)(x - 4)(x - 5)(x - 6)(x - 7)(x - 721/720)

Therefore, we can write:

(x² - 1)p(x) - x = (x - 2)(x - 3)(x - 4)(x - 5)(x - 6)(x - 7)(x - 721/720)

Therefore, solving for p(x):

p(x) = [ (x - 2)(x - 3)(x - 4)(x - 5)(x - 6)(x - 7)(x - 721/720) + x ] / (x² - 1)

Therefore, to find p(8), we can plug x = 8 into this expression.

So let's compute p(8):

First, compute the numerator:

N = (8 - 2)(8 - 3)(8 - 4)(8 - 5)(8 - 6)(8 - 7)(8 - 721/720) + 8

Let me compute each term:

(8 - 2) = 6

(8 - 3) = 5

(8 - 4) = 4

(8 - 5) = 3

(8 - 6) = 2

(8 - 7) = 1

(8 - 721/720) = 8 - 721/720. Let me convert 8 to 5760/720, so 5760/720 - 721/720 = (5760 - 721)/720 = 5039/720

Therefore, the product of the first six terms is 6*5*4*3*2*1 = 720

Then multiply by 5039/720:

720 * (5039/720) = 5039

Then add 8: N = 5039 + 8 = 5047

Then the denominator is (8² - 1) = 64 - 1 = 63

Therefore, p(8) = 5047 / 63

Wait, let me check that again. Wait, the numerator is [product + 8], which is 5039 + 8 = 5047? Wait, no. Wait, the product of the first six terms is 6*5*4*3*2*1 = 720, then multiplied by (8 - 721/720) which is 5039/720, so 720*(5039/720) = 5039. Then add x, which is 8? Wait, no. Wait, the original expression is:

N = [ (product) + x ] ?

Wait, no. Wait, the original expression is:

q(x) = (x² - 1)p(x) - x = product*(x - r)

Therefore, solving for p(x):

(x² - 1)p(x) = q(x) + x = product*(x - r) + x

Therefore, p(x) = [ product*(x - r) + x ] / (x² - 1)

Therefore, when x = 8, p(8) = [ product*(8 - r) + 8 ] / (8² - 1)

Wait, but earlier I said that product*(8 - r) is 720*(5039/720) = 5039, then adding 8? Wait, no. Wait, the product is (x - 2)...(x - 7) which at x=8 is 6*5*4*3*2*1 = 720, then multiplied by (x - r) which is (8 - r) = 5039/720. Therefore, the entire product term is 720*(5039/720) = 5039. Then adding x, which is 8? Wait, no. Wait, the formula is:

q(x) = product*(x - r) = (x² - 1)p(x) - x

Therefore, (x² - 1)p(x) = q(x) + x = product*(x - r) + x

Therefore, p(x) = [ product*(x - r) + x ] / (x² - 1)

Therefore, when x = 8, p(8) = [ product*(8 - r) + 8 ] / (8² - 1)

But product*(8 - r) is 720*(5039/720) = 5039, then adding 8 gives 5039 + 8 = 5047. Then divide by 63. So p(8) = 5047 / 63.

Wait, but let me check if that reduces. Let me compute 5047 divided by 63.

First, 63*80 = 5040. Therefore, 5047 - 5040 = 7. Therefore, 5047 / 63 = 80 + 7/63 = 80 + 1/9 = 80.111...

But the question says p(x) is a polynomial, so p(8) should be a rational number. But maybe 5047/63 can be simplified? Let me check if 5047 and 63 have common factors.

63 factors into 7*9. Let me check if 7 divides 5047. 7*721 = 5047? Wait, 7*700 = 4900, 7*21 = 147, so 4900 + 147 = 5047. Yes! So 5047 = 7*721. Therefore, 5047 / 63 = (7*721)/(7*9) ) = 721/9. Therefore, p(8) = 721/9.

Wait, that's better. So 721 divided by 9. Let me check 9*80 = 720, so 721/9 = 80 + 1/9, which is 80.111..., which matches the earlier decimal. So 721/9 is the simplified fraction.

But let me check my steps again to make sure I didn't make a mistake.

First, we had q(x) = (x² - 1)p(x) - x, which has roots at 2 through 7, so we wrote it as (x - 2)...(x - 7)(x - r). Then evaluated at x=1, found that q(1) = -1, which led us to find r = 721/720. Then, when computing p(8), we calculated the product (8 - 2)...(8 - 7) = 6! = 720, multiplied by (8 - r) = 8 - 721/720 = (5760 - 721)/720 = 5039/720. Then 720*(5039/720) = 5039. Then adding x, which is 8? Wait, no. Wait, the formula is [product*(x - r) + x] / (x² - 1). Wait, so product*(x - r) is 5039, then adding x which is 8? Wait, no, the formula is:

(x² - 1)p(x) = product*(x - r) + x. Therefore, p(x) = [product*(x - r) + x]/(x² - 1). Therefore, when x = 8, it's [product*(8 - r) + 8]/(8² - 1). So product*(8 - r) is 5039, then adding 8 gives 5039 + 8 = 5047. Then divide by 63. Then 5047 / 63 = 721/9. So that seems correct.

But let me check if 721/9 is the correct answer. Let me check if there's another way to verify.

Alternatively, maybe I can check with another value. For example, let me check if p(1) can be calculated? Wait, but we don't have p(1) given. However, since we have q(x) = (x² - 1)p(x) - x, then p(1) would be [q(1) + 1]/(1² - 1). But denominator is zero, so that's undefined. Which makes sense because p(x) is a polynomial, but when we plug x=1 into the original equation, we have division by zero. So that's okay.

Alternatively, maybe check for another value. Let me try x=0. Wait, but we don't have p(0) given. Maybe not helpful.

Alternatively, maybe check if the answer 721/9 makes sense. Let me think. Let me compute 721 divided by 9. 9*80 = 720, so 721/9 = 80 + 1/9 ≈ 80.111...

Alternatively, maybe there's a different approach. Let me think.

Another approach: Since p(x) is a degree 5 polynomial, and we know its values at 6 points, maybe we can use finite differences or something else? But that might be more complicated.

Alternatively, consider that the function f(x) = x/(x² - 1) can be written as f(x) = 1/(x - 1) - 1/(x + 1). Let me check:

1/(x - 1) - 1/(x + 1) = [ (x + 1) - (x - 1) ] / (x² - 1) ) = [2]/(x² - 1). Wait, that's 2/(x² - 1). So f(x) = x/(x² - 1) is not directly that. But maybe there's a relation.

Alternatively, maybe consider that f(x) = [x/(x² - 1)] = [1/(x - 1)] - [1/(x + 1)] / 2? Wait, let me check:

Let me compute [1/(x - 1) - 1/(x + 1)] / 2 = [ (x + 1 - x + 1) / (x² - 1) ] / 2 = [2 / (x² - 1)] / 2 = 1/(x² - 1). So that's different from f(x). So f(x) is x/(x² - 1). Maybe not helpful directly.

Alternatively, think about constructing a polynomial that interpolates these points. But with Lagrange interpolation, but with 6 points, it's going to be tedious. However, maybe there's a pattern.

Alternatively, think about the fact that p(x) - f(x) has roots at 2 through 7, but f(x) isn't a polynomial. However, we already did that approach earlier, which led us to the answer. So maybe that's the correct way.

Alternatively, maybe check if 721/9 is correct by plugging into the equation. Let me check if with p(8) = 721/9, then (8² - 1)p(8) - 8 should equal q(8) = (8 - 2)...(8 - 7)(8 - r). Let me compute both sides.

Left side: (64 - 1)*(721/9) - 8 = 63*(721/9) - 8 = 7*721 - 8 = 5047 - 8 = 5039.

Right side: product*(8 - r) = 720*(8 - 721/720) = 720*( (5760 - 721)/720 ) = 5039. Which matches. Therefore, that checks out. Therefore, p(8) = 721/9 is correct.

Therefore, the answer is 721/9. Let me check if that reduces further. 721 divided by... Let me check if 721 is divisible by any number. 721 ÷ 7 = 103, since 7*100=700, 7*3=21, so 7*103=721. Therefore, 721 = 7*103. Then 9 is 3². So no common factors. Therefore, 721/9 is reduced.

Therefore, the answer is 721/9. So boxed, it's \boxed{\dfrac{721}{9}}.

**Final Answer**
\boxed{\dfrac{721}{9}}

To solve the problem, we are given a polynomial $ p(x) $ of degree 5 such that:

$$
p(n) = \frac{n}{n^2 - 1} \quad \text{for } n = 2, 3, 4, 5, 6, 7
$$

We are asked to find $ p(8) $.

---

### Step 1: Define a Related Polynomial

Let us define a new polynomial:

$$
q(x) = (x^2 - 1)p(x) - x
$$

This is a polynomial of degree 7, since $ p(x) $ is degree 5 and $ x^2 - 1 $ is degree 2. The subtraction of $ x $ does not change the degree.

Now, observe that for $ x = 2, 3, 4, 5, 6, 7 $, we have:

$$
q(n) = (n^2 - 1)p(n) - n = (n^2 - 1)\cdot\frac{n}{n^2 - 1} - n = n - n = 0
$$

So, $ q(x) $ has roots at $ x = 2, 3, 4, 5, 6, 7 $. Therefore, we can write:

$$
q(x) = (x - 2)(x - 3)(x - 4)(x - 5)(x - 6)(x - 7)(x - r)
$$

for some unknown root $ r $.

---

### Step 2: Determine the Missing Root

We can find $ r $ by evaluating $ q(x) $ at a point not in the given set. A natural choice is $ x = 1 $, since $ x^2 - 1 = 0 $ at $ x = 1 $, and we can compute $ q(1) $ in two ways.

From the definition:

$$
q(1) = (1^2 - 1)p(1) - 1 = 0 \cdot p(1) - 1 = -1
$$

From the factored form:

$$
q(1) = (1 - 2)(1 - 3)(1 - 4)(1 - 5)(1 - 6)(1 - 7)(1 - r)
$$

Compute the product:

$$
(-1)(-2)(-3)(-4)(-5)(-6)(1 - r) = 720(1 - r)
$$

So:

$$
720(1 - r) = -1 \Rightarrow 1 - r = -\frac{1}{720} \Rightarrow r = 1 + \frac{1}{720} = \frac{721}{720}
$$

---

### Step 3: Use the Polynomial to Find $ p(8) $

We now have:

$$
q(x) = (x - 2)(x - 3)(x - 4)(x - 5)(x - 6)(x - 7)\left(x - \frac{721}{720}\right)
$$

Recall that:

$$
q(x) = (x^2 - 1)p(x) - x \Rightarrow p(x) = \frac{q(x) + x}{x^2 - 1}
$$

So, to find $ p(8) $, compute:

$$
p(8) = \frac{q(8) + 8}{8^2 - 1} = \frac{q(8) + 8}{63}
$$

Now compute $ q(8) $:

$$
q(8) = (8 - 2)(8 - 3)(8 - 4)(8 - 5)(8 - 6)(8 - 7)\left(8 - \frac{721}{720}\right)
$$

Compute each term:

- $ 8 - 2 = 6 $
- $ 8 - 3 = 5 $
- $ 8 - 4 = 4 $
- $ 8 - 5 = 3 $
- $ 8 - 6 = 2 $
- $ 8 - 7 = 1 $
- $ 8 - \frac{721}{720} = \frac{5760 - 721}{720} = \frac{5039}{720} $

So:

$$
q(8) = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 \cdot \frac{5039}{720} = 720 \cdot \frac{5039}{720} = 5039
$$

Then:

$$
p(8) = \frac{5039 + 8}{63} = \frac{5047}{63}
$$

Now simplify:

$$
\frac{5047}{63} = \frac{7 \cdot 721}{7 \cdot 9} = \frac{721}{9}
$$

---

### Final Answer

$$
\boxed{\dfrac{721}{9}}
$$

Solution:
Let $q(x) = (x^2 - 1) p(x) - x.$  Then $q(x)$ has degree 7, and $q(n) = 0$ for $n = 2$, 3, 4, $\dots,$ 7, so
\[q(x) = (ax + b)(x - 2)(x - 3) \dotsm (x - 7)\]for some constants $a$ and $b.$

We know that $q(1) = (1^2 - 1)p(1) - 1 = -1.$  Setting $x = 1$ in the equation above, we get
\[q(1) = 720(a + b),\]so $a + b = -\frac{1}{720}.$

We also know that $q(-1) = ((-1)^2 - 1)p(-1) + 1 = 1.$  Setting $x = -1$ in the equation above, we get
\[q(-1) = 20160(-a + b),\]so $-a + b = \frac{1}{20160}.$  Solving for $a$ and $b,$ we find $a = -\frac{29}{40320}$ and $b = -\frac{3}{4480}.$  Hence,
\begin{align*}
q(x) &= \left( -\frac{29}{40320} x - \frac{3}{4480} \right) (x - 2)(x - 3) \dotsm (x - 7) \\
&= -\frac{(29x + 27)(x - 2)(x - 3) \dotsm (x - 7)}{40320}.
\end{align*}In particular,
\[q(8) = -\frac{(29 \cdot 8 + 27)(6)(5) \dotsm (1)}{40320} = -\frac{37}{8},\]so
\[p(8) = \frac{q(8) + 8}{8^2 - 1} = \boxed{\frac{3}{56}}.\]


Explanation:
Okay, so I need to find the sum of the proper divisors of the sum of the proper divisors of 284. Let me break this down step by step. First, I should figure out what the proper divisors of 284 are, then sum them up, and finally find the sum of the proper divisors of that resulting number.

Starting with the first part: finding the proper divisors of 284. Proper divisors are all the divisors of a number except the number itself. So, I need to list all numbers that divide 284 evenly without leaving a remainder, and then exclude 284 itself.

Let me recall how to find divisors. One way is to factorize the number into its prime factors and then use that to find all divisors. Let me try that.

First, factorizing 284. Let me divide by small primes.

284 divided by 2 is 142. Then 142 divided by 2 is 71. Now, 71 is a prime number, right? Let me check. 71 divided by 2 is not whole, 3? 3*23 is 69, 3*24 is 72, so no. 5? Doesn't end with 0 or 5. 7? 7*10 is 70, 7*11 is 77, so no. 11? 11*6 is 66, 11*7 is 77. So yes, 71 is prime.

So the prime factorization of 284 is 2² × 71¹.

Now, the formula for finding the number of divisors is (exponent + 1) for each prime factor multiplied together. But here, I need the actual divisors, not just the count.

Alternatively, since I know the prime factors, I can generate all divisors by taking combinations of the exponents.

For 2² × 71¹, the exponents for 2 can be 0, 1, or 2, and for 71 can be 0 or 1.

So the divisors are:

- 2⁰ × 71⁰ = 1
- 2¹ × 71⁰ = 2
- 2² × 71⁰ = 4
- 2⁰ × 71¹ = 71
- 2¹ × 71¹ = 2×71 = 142
- 2² × 71¹ = 4×71 = 284

But since proper divisors exclude the number itself, we remove 284. Therefore, the proper divisors of 284 are 1, 2, 4, 71, and 142.

Let me check if these are correct. Let me verify by dividing 284 by each of these numbers:

- 284 ÷ 1 = 284, which is okay.
- 284 ÷ 2 = 142, which is an integer.
- 284 ÷ 4 = 71, which is an integer.
- 284 ÷ 71 = 4, which is an integer.
- 284 ÷ 142 = 2, which is an integer.

So yes, those are all the proper divisors.

Now, the next step is to find the sum of these proper divisors. Let me add them up:

1 + 2 + 4 + 71 + 142.

Let me compute that step by step:

1 + 2 = 3

3 + 4 = 7

7 + 71 = 78

78 + 142 = 220

So the sum of the proper divisors of 284 is 220.

Now, the problem asks for the sum of the proper divisors of this sum, which is 220. So now I need to find the proper divisors of 220 and then add them up.

Again, let me start by finding the proper divisors of 220.

First, let me factorize 220.

220 divided by 2 is 110. Divided by 2 again is 55. Then 55 divided by 5 is 11. And 11 is prime.

So the prime factorization of 220 is 2² × 5¹ × 11¹.

To find all divisors, I can use the exponents. For each prime factor, exponents go from 0 up to their respective exponents.

So for 2², exponents 0,1,2; for 5¹, exponents 0,1; for 11¹, exponents 0,1.

Therefore, the divisors are all combinations:

- 2⁰ × 5⁰ × 11⁰ = 1
- 2¹ × 5⁰ × 11⁰ = 2
- 2² × 5⁰ × 11⁰ = 4
- 2⁰ × 5¹ × 11⁰ = 5
- 2¹ × 5¹ × 11⁰ = 10
- 2² × 5¹ × 11⁰ = 20
- 2⁰ × 5⁰ × 11¹ = 11
- 2¹ × 5⁰ × 11¹ = 22
- 2² × 5⁰ × 11¹ = 44
- 2⁰ × 5¹ × 11¹ = 55
- 2¹ × 5¹ × 11¹ = 110
- 2² × 5¹ × 11¹ = 220

So the divisors of 220 are 1, 2, 4, 5, 10, 20, 11, 22, 44, 55, 110, 220.

Proper divisors exclude 220, so they are 1, 2, 4, 5, 10, 20, 11, 22, 44, 55, 110.

Now, let me sum these up.

Let me list them again:

1, 2, 4, 5, 10, 20, 11, 22, 44, 55, 110.

Adding them step by step:

Start with 1 + 2 = 3

3 + 4 = 7

7 + 5 = 12

12 + 10 = 22

22 + 20 = 42

42 + 11 = 53

53 + 22 = 75

75 + 44 = 119

119 + 55 = 174

174 + 110 = 284

Wait, so the sum of the proper divisors of 220 is 284?

But hold on, that seems interesting. Because 284 is the original number we started with.

But let me check my addition again because that seems like a possible mistake. Let me verify each step again:

Starting with 1, 2, 4, 5, 10, 20, 11, 22, 44, 55, 110.

Let me group them to make addition easier:

First, group smaller numbers:

1 + 2 + 4 + 5 = 12

Then 10 + 20 = 30

Then 11 + 22 = 33

Then 44 + 55 = 99

Then the remaining is 110.

So adding those groups:

12 + 30 = 42

42 + 33 = 75

75 + 99 = 174

174 + 110 = 284.

Yes, same result. So the sum of the proper divisors of 220 is indeed 284.

Therefore, the answer to the original question, which is the sum of the proper divisors of the sum of the proper divisors of 284, is 284.

But let me just confirm once more that I didn't make a mistake in the steps.

First, proper divisors of 284: 1, 2, 4, 71, 142. Sum is 1+2+4+71+142=220. Then proper divisors of 220 sum to 284.

But wait, 284 and 220 are known as amicable numbers. Amicable numbers are pairs of numbers where each is the sum of the proper divisors of the other. So 220 and 284 form an amicable pair.

Therefore, this seems correct.

But just to be thorough, let me check the sum of proper divisors of 220 again. Maybe I missed a divisor or added incorrectly.

List of proper divisors of 220: 1, 2, 4, 5, 10, 20, 11, 22, 44, 55, 110.

Let me check if there are any other divisors.

For example, 220 divided by 3 is not an integer. Divided by 6? 220 ÷ 6 is about 36.666... Not integer. Divided by 7? 220 ÷ 7 ≈ 31.428... Not integer. Divided by 8? 220 ÷ 8 = 27.5. Not integer. Divided by 9? 220 ÷ 9 ≈ 24.444... Not integer. Divided by 12? 220 ÷ 12 ≈ 18.333... Not integer. Divided by 13? 220 ÷ 13 ≈ 16.923... Not integer. Divided by 14? 220 ÷ 14 ≈ 15.714... Not integer. Divided by 15? 220 ÷ 15 ≈ 14.666... Not integer. Divided by 16? 220 ÷ 16 = 13.75. Not integer. Divided by 17? 220 ÷ 17 ≈ 12.94... Not integer. Divided by 18? 220 ÷ 18 ≈ 12.222... Not integer. Divided by 19? 220 ÷ 19 ≈ 11.578... Not integer. Divided by 20? 220 ÷ 20 = 11, which is already in the list.

So, seems like all divisors are accounted for.

Therefore, the sum is indeed 284.

Therefore, the answer is 284.

But let me check once more the sum of the proper divisors of 220. Maybe I can use another method.

There's a formula for the sum of divisors function, σ(n), which includes the number itself. Then the sum of proper divisors is σ(n) - n.

So if I can compute σ(220) and subtract 220, that should give me the sum of proper divisors.

Given that 220 factors into 2² × 5 × 11, the sum of divisors function σ(n) is (1 + 2 + 2²) × (1 + 5) × (1 + 11).

Calculating each part:

For 2²: 1 + 2 + 4 = 7

For 5: 1 + 5 = 6

For 11: 1 + 11 = 12

Multiply them together: 7 × 6 × 12.

First, 7 × 6 = 42. Then 42 × 12. Let me compute that:

42 × 10 = 420, 42 × 2 = 84, so total is 420 + 84 = 504.

Therefore, σ(220) = 504.

Therefore, the sum of proper divisors is 504 - 220 = 284.

Yes! That matches my previous result. So that confirms that the sum of proper divisors of 220 is indeed 284.

Therefore, the answer is 284.

But just to recap all steps once again to make sure I didn't skip anything:

1. Find proper divisors of 284: 1, 2, 4, 71, 142. Sum is 220.

2. Find proper divisors of 220: 1, 2, 4, 5, 10, 20, 11, 22, 44, 55, 110. Sum is 284.

Therefore, the answer is 284.

Alternatively, since 284 is the original number, and the problem is about the sum of proper divisors of the sum of proper divisors of 284, which loops back to 284.

Therefore, I think this is correct.

I don't see any mistakes in my reasoning. The key was recognizing that 220 and 284 are amicable numbers, but even without that knowledge, verifying through multiple methods (direct addition and using the sum of divisors function) confirms the result.

So I feel confident that the answer is 284.

**Final Answer**
\boxed{284}

To solve the problem, we are asked to find the **sum of the proper divisors of the sum of the proper divisors of 284**. Let's break this down step by step.

---

### Step 1: Find the Proper Divisors of 284

A **proper divisor** of a number is a positive divisor that is **less than** the number itself.

We begin by **factoring 284**:

$$
284 = 2^2 \times 71
$$

From this, we can list all the **divisors** of 284:

$$
1, 2, 4, 71, 142, 284
$$

Excluding 284 (since it's the number itself), the **proper divisors** of 284 are:

$$
1, 2, 4, 71, 142
$$

Now, compute their **sum**:

$$
1 + 2 + 4 + 71 + 142 = 220
$$

---

### Step 2: Find the Proper Divisors of 220

Now we need to find the **proper divisors of 220**. First, factorize 220:

$$
220 = 2^2 \times 5 \times 11
$$

Using this, we can list all the **divisors** of 220:

$$
1, 2, 4, 5, 10, 20, 11, 22, 44, 55, 110, 220
$$

Excluding 220, the **proper divisors** are:

$$
1, 2, 4, 5, 10, 20, 11, 22, 44, 55, 110
$$

Now, compute their **sum**:

$$
1 + 2 + 4 + 5 + 10 + 20 + 11 + 22 + 44 + 55 + 110 = 284
$$

---

### Final Result

Thus, the **sum of the proper divisors of the sum of the proper divisors of 284** is:

$$
\boxed{284}
$$

[Assistant]
Explanation:
Okay, so I need to find the sum of the proper divisors of the sum of the proper divisors of 284. Let me break this down step by step. First, I should figure out what the proper divisors of 284 are, then sum them up, and finally find the sum of the proper divisors of that resulting number.

Starting with the first part: finding the proper divisors of 284. Proper divisors are all the divisors of a number except the number itself. So, I need to list all numbers that divide 284 evenly without leaving a remainder, and then exclude 284 itself.

Let me recall how to find divisors. One way is to factorize the number into its prime factors and then use that to find all divisors. Let me try that.

First, factorizing 284. Let me divide by small primes.

284 divided by 2 is 142. Then 142 divided by 2 is 71. Now, 71 is a prime number, right? Let me check. 71 divided by 2 is not whole, 3? 3*23 is 69, 3*24 is 72, so no. 5? Doesn't end with 0 or 5. 7? 7*10 is 70, 7*11 is 77, so no. 11? 11*6 is 66, 11*7 is 77. So yes, 71 is prime.

So the prime factorization of 284 is 2² × 71¹.

Now, the formula for finding the number of divisors is (exponent + 1) for each prime factor multiplied together. But here, I need the actual divisors, not just the count.

Alternatively, since I know the prime factors, I can generate all divisors by taking combinations of the exponents.

For 2² × 71¹, the exponents for 2 can be 0, 1, or 2, and for 71 can be 0 or 1.

So the divisors are:

- 2⁰ × 71⁰ = 1
- 2¹ × 71⁰ = 2
- 2² × 71⁰ = 4
- 2⁰ × 71¹ = 71
- 2¹ × 71¹ = 2×71 = 142
- 2² × 71¹ = 4×71 = 284

But since proper divisors exclude the number itself, we remove 284. Therefore, the proper divisors of 284 are 1, 2, 4, 71, and 142.

Let me check if these are correct. Let me verify by dividing 284 by each of these numbers:

- 284 ÷ 1 = 284, which is okay.
- 284 ÷ 2 = 142, which is an integer.
- 284 ÷ 4 = 71, which is an integer.
- 284 ÷ 71 = 4, which is an integer.
- 284 ÷ 142 = 2, which is an integer.

So yes, those are all the proper divisors.

Now, the next step is to find the sum of these proper divisors. Let me add them up:

1 + 2 + 4 + 71 + 142.

Let me compute that step by step:

1 + 2 = 3

3 + 4 = 7

7 + 71 = 78

78 + 142 = 220

So the sum of the proper divisors of 284 is 220.

Now, the problem asks for the sum of the proper divisors of this sum, which is 220. So now I need to find the proper divisors of 220 and then add them up.

Again, let me start by finding the proper divisors of 220.

First, let me factorize 220.

220 divided by 2 is 110. Divided by 2 again is 55. Then 55 divided by 5 is 11. And 11 is prime.

So the prime factorization of 220 is 2² × 5¹ × 11¹.

To find all divisors, I can use the exponents. For each prime factor, exponents go from 0 up to their respective exponents.

So for 2², exponents 0,1,2; for 5¹, exponents 0,1; for 11¹, exponents 0,1.

Therefore, the divisors are all combinations:

- 2⁰ × 5⁰ × 11⁰ = 1
- 2¹ × 5⁰ × 11⁰ = 2
- 2² × 5⁰ × 11⁰ = 4
- 2⁰ × 5¹ × 11⁰ = 5
- 2¹ × 5¹ × 11⁰ = 10
- 2² × 5¹ × 11⁰ = 20
- 2⁰ × 5⁰ × 11¹ = 11
- 2¹ × 5⁰ × 11¹ = 22
- 2² × 5⁰ × 11¹ = 44
- 2⁰ × 5¹ × 11¹ = 55
- 2¹ × 5¹ × 11¹ = 110
- 2² × 5¹ × 11¹ = 220

So the divisors of 220 are 1, 2, 4, 5, 10, 20, 11, 22, 44, 55, 110, 220.

Proper divisors exclude 220, so they are 1, 2, 4, 5, 10, 20, 11, 22, 44, 55, 110.

Now, let me sum these up.

Let me list them again:

1, 2, 4, 5, 10, 20, 11, 22, 44, 55, 110.

Adding them step by step:

Start with 1 + 2 = 3

3 + 4 = 7

7 + 5 = 12

12 + 10 = 22

22 + 20 = 42

42 + 11 = 53

53 + 22 = 75

75 + 44 = 119

119 + 55 = 174

174 + 110 = 284

Wait, so the sum of the proper divisors of 220 is 284?

But hold on, that seems interesting. Because 284 is the original number we started with.

But let me check my addition again because that seems like a possible mistake. Let me verify each step again:

Starting with 1, 2, 4, 5, 10, 20, 11, 22, 44, 55, 110.

Let me group them to make addition easier:

First, group smaller numbers:

1 + 2 + 4 + 5 = 12

Then 10 + 20 = 30

Then 11 + 22 = 33

Then 44 + 55 = 99

Then the remaining is 110.

So adding those groups:

12 + 30 = 42

42 + 33 = 75

75 + 99 = 174

174 + 110 = 284.

Yes, same result. So the sum of the proper divisors of 220 is indeed 284.

Therefore, the answer to the original question, which is the sum of the proper divisors of the sum of the proper divisors of 284, is 284.

But let me just confirm once more that I didn't make a mistake in the steps.

First, proper divisors of 284: 1, 2, 4, 71, 142. Sum is 1+2+4+71+142=220. Then proper divisors of 220 sum to 284.

But wait, 284 and 220 are known as amicable numbers. Amicable numbers are pairs of numbers where each is the sum of the proper divisors of the other. So 220 and 284 form an amicable pair.

Therefore, this seems correct.

But just to be thorough, let me check the sum of proper divisors of 220 again. Maybe I missed a divisor or added incorrectly.

List of proper divisors of 220: 1, 2, 4, 5, 10, 20, 11, 22, 44, 55, 110.

Let me check if there are any other divisors.

For example, 220 divided by 3 is not an integer. Divided by 6? 220 ÷ 6 is about 36.666... Not integer. Divided by 7? 220 ÷ 7 ≈ 31.428... Not integer. Divided by 8? 220 ÷ 8 = 27.5. Not integer. Divided by 9? 220 ÷ 9 ≈ 24.444... Not integer. Divided by 12? 220 ÷ 12 ≈ 18.333... Not integer. Divided by 13? 220 ÷ 13 ≈ 16.923... Not integer. Divided by 14? 220 ÷ 14 ≈ 15.714... Not integer. Divided by 15? 220 ÷ 15 ≈ 14.666... Not integer. Divided by 16? 220 ÷ 16 = 13.75. Not integer. Divided by 17? 220 ÷ 17 ≈ 12.94... Not integer. Divided by 18? 220 ÷ 18 ≈ 12.222... Not integer. Divided by 19? 220 ÷ 19 ≈ 11.578... Not integer. Divided by 20? 220 ÷ 20 = 11, which is already in the list.

So, seems like all divisors are accounted for.

Therefore, the sum is indeed 284.

Therefore, the answer is 284.

But let me check once more the sum of the proper divisors of 220. Maybe I can use another method.

There's a formula for the sum of divisors function, σ(n), which includes the number itself. Then the sum of proper divisors is σ(n) - n.

So if I can compute σ(220) and subtract 220, that should give me the sum of proper divisors.

Given that 220 factors into 2² × 5 × 11, the sum of divisors function σ(n) is (1 + 2 + 2²) × (1 + 5) × (1 + 11).

Calculating each part:

For 2²: 1 + 2 + 4 = 7

For 5: 1 + 5 = 6

For 11: 1 + 11 = 12

Multiply them together: 7 × 6 × 12.

First, 7 × 6 = 42. Then 42 × 12. Let me compute that:

42 × 10 = 420, 42 × 2 = 84, so total is 420 + 84 = 504.

Therefore, σ(220) = 504.

Therefore, the sum of proper divisors is 504 - 220 = 284.

Yes! That matches my previous result. So that confirms that the sum of proper divisors of 220 is indeed 284.

Therefore, the answer is 284.

But just to recap all steps once again to make sure I didn't skip anything:

1. Find proper divisors of 284: 1, 2, 4, 71, 142. Sum is 220.

2. Find proper divisors of 220: 1, 2, 4, 5, 10, 20, 11, 22, 44, 55, 110. Sum is 284.

Therefore, the answer is 284.

Alternatively, since 284 is the original number, and the problem is about the sum of proper divisors of the sum of proper divisors of 284, which loops back to 284.

Therefore, I think this is correct.

I don't see any mistakes in my reasoning. The key was recognizing that 220 and 284 are amicable numbers, but even without that knowledge, verifying through multiple methods (direct addition and using the sum of divisors function) confirms the result.

So I feel confident that the answer is 284.

**Final Answer**
\boxed{284}

To solve the problem, we are asked to find the **sum of the proper divisors of the sum of the proper divisors of 284**. Let's break this down step by step.

---

### Step 1: Find the Proper Divisors of 284

A **proper divisor** of a number is a positive divisor that is **less than** the number itself.

We begin by **factoring 284**:

$$
284 = 2^2 \times 71
$$

From this, we can list all the **divisors** of 284:

$$
1, 2, 4, 71, 142, 284
$$

Excluding 284 (since it's the number itself), the **proper divisors** of 284 are:

$$
1, 2, 4, 71, 142
$$

Now, compute their **sum**:

$$
1 + 2 + 4 + 71 + 142 = 220
$$

---

### Step 2: Find the Proper Divisors of 220

Now we need to find the **proper divisors of 220**. First, factorize 220:

$$
220 = 2^2 \times 5 \times 11
$$

Using this, we can list all the **divisors** of 220:

$$
1, 2, 4, 5, 10, 20, 11, 22, 44, 55, 110, 220
$$

Excluding 220, the **proper divisors** are:

$$
1, 2, 4, 5, 10, 20, 11, 22, 44, 55, 110
$$

Now, compute their **sum**:

$$
1 + 2 + 4 + 5 + 10 + 20 + 11 + 22 + 44 + 55 + 110 = 284
$$

---

### Final Result

Thus, the **sum of the proper divisors of the sum of the proper divisors of 284** is:

$$
\boxed{284}
$$

Solution:
Prime factorize $284=2^2\cdot71$. The sum of the proper divisors of $284$ is
\begin{align*}
1+2+2^2+71+2 \cdot 71 &= (1+2+2^2)(1+71)-284 \\
&= 220 \\
&= 2^2\cdot5\cdot11.
\end{align*}Here we have used the observation that multiplying out $(1+2+2^2)(1+71)$ by distributing yields an expression which is the sum of all $6$ factors of $284.$ Applying this observation again, we find that the sum of the proper divisors of $220$ is $$(1+2+2^2)(1+5)(1+11)-220=7\cdot 6\cdot 12-220=\boxed{284}.$$


Explanation:
Okay, so I need to find the complex number w, which is the result of rotating z around c by π/4 radians counter-clockwise. Let me recall how to perform such rotations in the complex plane.

First, I remember that rotating a complex number around another point involves translating the complex plane so that the center of rotation becomes the origin, performing the rotation, and then translating back. So, the general formula for rotating a point z around a point c by an angle θ is:

w = (z - c) * e^{iθ} + c

Yes, that seems right. Let me verify. If I subtract c from z, that moves the center of rotation to the origin. Then multiplying by e^{iθ} rotates it by θ, and then adding c brings it back to the original position. So, that formula should work.

Given that, let me write down the given values.

z is given as 2 + √2 - (3 + 3√2)i. Let me write that in standard form. The real part is 2 + √2, and the imaginary part is -(3 + 3√2). So, z = (2 + √2) - (3 + 3√2)i.

c is given as 2 - 3i. So, c = 2 - 3i.

The angle θ is π/4 radians, which is 45 degrees. The rotation is counter-clockwise, so the rotation factor is e^{iπ/4}.

So, applying the formula:

w = (z - c) * e^{iπ/4} + c

First, I need to compute z - c. Let me compute that.

z - c = [ (2 + √2) - (3 + 3√2)i ] - [2 - 3i]

Let me subtract the real parts and the imaginary parts separately.

Real part: (2 + √2) - 2 = √2

Imaginary part: [ - (3 + 3√2)i ] - (-3i) = -3 - 3√2 + 3 = -3√2 i

Wait, let me check that again. The imaginary part of z is -(3 + 3√2)i, and the imaginary part of c is -3i. So when subtracting c, which is subtracting 2 - 3i, so subtracting the real part and subtracting the imaginary part. Wait, actually, when you subtract c from z, it's z - c = (real part of z - real part of c) + (imaginary part of z - imaginary part of c)i.

So, real part: (2 + √2) - 2 = √2. Correct.

Imaginary part: [ - (3 + 3√2) ] - (-3) = -3 - 3√2 + 3 = -3√2. So the imaginary part is -3√2. Therefore, z - c = √2 - 3√2 i.

So z - c = √2(1 - 3i). Let me write that as √2*(1 - 3i).

Now, I need to multiply this by e^{iπ/4}. Let me recall that e^{iπ/4} is cos(π/4) + i sin(π/4) = (√2/2) + i(√2/2).

Alternatively, I can write e^{iπ/4} as (1 + i)/√2. Let me confirm:

(1 + i)/√2 has magnitude √( (1/√2)^2 + (1/√2)^2 ) = √(1/2 + 1/2) = √1 = 1, and angle arctan( (1/√2)/(1/√2) ) = arctan(1) = π/4. Yes, correct. So e^{iπ/4} = (1 + i)/√2.

Therefore, multiplying (z - c) by e^{iπ/4} is equivalent to multiplying √2*(1 - 3i) by (1 + i)/√2. The √2 cancels with 1/√2, so we get (1 - 3i)(1 + i).

Let me compute that:

(1 - 3i)(1 + i) = 1*(1) + 1*(i) - 3i*(1) - 3i*(i) = 1 + i - 3i - 3i².

Simplify:

1 + i - 3i - 3(-1) = 1 - 2i + 3 = 4 - 2i.

So, the product (z - c)*e^{iπ/4} is 4 - 2i. Then, adding back c, which is 2 - 3i, gives:

w = (4 - 2i) + (2 - 3i) = (4 + 2) + (-2i - 3i) = 6 - 5i.

Wait, so is that the answer? Let me check my steps again to make sure I didn't make a mistake.

First, z - c:

z = 2 + √2 - (3 + 3√2)i

c = 2 - 3i

So z - c = (2 + √2 - 2) + [ - (3 + 3√2)i - (-3i) ] = √2 + [ -3 - 3√2 + 3 ]i = √2 - 3√2 i. Correct. So that's √2(1 - 3i). Then multiplying by e^{iπ/4} which is (1 + i)/√2. So √2*(1 - 3i)*(1 + i)/√2 = (1 - 3i)(1 + i). Then expanding that:

1*1 + 1*i - 3i*1 - 3i*i = 1 + i - 3i - 3i². Since i² = -1, so -3i² = 3. Therefore, 1 + 3 + i - 3i = 4 - 2i. Correct. Then adding c, which is 2 - 3i. So 4 - 2i + 2 - 3i = 6 - 5i.

Wait, but let me check if the rotation direction is correct. The problem says counter-clockwise, which is the standard direction for positive angles. Since we used e^{iπ/4}, which is a counter-clockwise rotation, that's correct. So seems okay.

But let me check with another approach. Maybe using rotation matrices?

Alternatively, since complex numbers can be represented as vectors, and rotation can be done with rotation matrices. Let me try that approach to verify.

First, z - c is the vector from c to z. So, if I consider z - c as a complex number, which we found to be √2 - 3√2 i. Let me write that as a vector ( √2, -3√2 ). Then, rotating this vector by π/4 counter-clockwise. The rotation matrix for π/4 is:

[ cos(π/4)  -sin(π/4) ]
[ sin(π/4)   cos(π/4) ]

Which is:

[ √2/2   -√2/2 ]
[ √2/2    √2/2 ]

So multiplying this matrix by the vector ( √2, -3√2 ):

First component: (√2/2)(√2) + (-√2/2)(-3√2)

Second component: (√2/2)(√2) + (√2/2)(-3√2)

Let me compute first component:

(√2/2)(√2) = (2)/2 = 1

(-√2/2)(-3√2) = (3*(√2)^2)/2 = 3*2/2 = 3

So total first component: 1 + 3 = 4

Second component:

(√2/2)(√2) = 1

(√2/2)(-3√2) = -3*(√2)^2 / 2 = -3*2 / 2 = -3

So total second component: 1 - 3 = -2

Therefore, the rotated vector is (4, -2), which corresponds to the complex number 4 - 2i. Then adding back c, which is (2, -3), so adding 2 - 3i gives (4 + 2, -2 + (-3)) = (6, -5), which is 6 - 5i. So same result. Therefore, seems correct.

But let me check once more with another method. Maybe converting to polar form?

But z - c is √2 - 3√2 i. Let me write that as √2(1 - 3i). Let me find its magnitude and angle.

Magnitude: |√2(1 - 3i)| = √2 * |1 - 3i| = √2 * √(1^2 + (-3)^2) = √2 * √(1 + 9) = √2 * √10 = √20 = 2√5.

Angle: arctangent of (-3)/1 = arctan(-3). Since the complex number is in the fourth quadrant (positive real, negative imaginary). So angle θ = -arctan(3) or equivalently 2π - arctan(3).

But when we rotate by π/4, the new angle will be θ + π/4. Then, converting back to rectangular form. However, this might be more complicated, but let me try.

Alternatively, since we already have the result via two methods, maybe it's safe. But let me check if the answer is 6 - 5i. Wait, but let me check with the Asymptote code given. The Asymptote code defines Z as (2 + sqrt(2), -3 - 3*sqrt(2)), which is the coordinates of z. Let me check if that matches with z.

Original z is 2 + sqrt(2) - (3 + 3 sqrt(2))i. So in coordinates, that would be (real part, imaginary part) = (2 + sqrt(2), - (3 + 3 sqrt(2))). Which is (2 + sqrt(2), -3 - 3 sqrt(2)). So yes, that's what the Asymptote code says. Then c is (2, -3). Then W is rotate(45, C)*(Z). So the Asymptote code is rotating Z around C by 45 degrees. So according to the Asymptote code, the answer should be the coordinates of W. Let me compute coordinates of W.

If my answer is 6 - 5i, then coordinates are (6, -5). Let me check if that's consistent with the rotation.

Alternatively, maybe I can compute coordinates step by step. Let me do that.

Original point Z is (2 + √2, -3 - 3√2). Center C is (2, -3). So vector from C to Z is ( (2 + √2 - 2), (-3 - 3√2 - (-3)) ) = (√2, -3√2). Which is same as before. Rotating this vector by π/4 gives (4, -2) as before. Then adding back C's coordinates (2, -3) gives (4 + 2, -2 + (-3)) = (6, -5). So yes, coordinates (6, -5), which is 6 - 5i. So that seems correct.

But let me check if the Asymptote code's drawing is consistent. The Asymptote code draws Z--C--W. So from Z to C to W. So if Z is (2 + sqrt(2), -3 - 3 sqrt(2)), C is (2, -3), and W is (6, -5). Let me check the distance from Z to C and from C to W. Wait, but since it's a rotation, the distance from Z to C should be same as from W to C. Let me check.

Distance from Z to C: sqrt( (√2)^2 + (-3√2)^2 ) = sqrt( 2 + 9*2 ) = sqrt(2 + 18) = sqrt(20) = 2√5.

Distance from W to C: sqrt( (6 - 2)^2 + (-5 - (-3))^2 ) = sqrt( (4)^2 + (-2)^2 ) = sqrt(16 + 4) = sqrt(20) = 2√5. So same distance. Good. Also, the angle between ZC and WC should be π/4. Let me check the angle.

But maybe that's too involved. Alternatively, since all methods give me 6 - 5i, and the Asymptote code seems to confirm the coordinates, I think that is correct.

But just to be thorough, let me check once again the multiplication step.

(z - c) = √2 - 3√2 i. Multiply by e^{iπ/4} = (1 + i)/√2.

So:

(√2 - 3√2 i) * (1 + i)/√2 = [√2(1 - 3i)] * (1 + i)/√2 = (1 - 3i)(1 + i) = 1*(1) + 1*(i) - 3i*(1) - 3i*(i) = 1 + i - 3i - 3i² = 1 - 2i + 3 = 4 - 2i. Correct. Then adding c = 2 - 3i gives 4 - 2i + 2 - 3i = 6 - 5i.

Yes, that's correct. Therefore, I think the answer is 6 - 5i.

But just to make sure, let me try another approach. Let me represent all numbers as complex numbers and perform the rotation step by step.

Let me write z = 2 + √2 - (3 + 3√2)i. Let me write that as z = (2 + √2) + (-3 - 3√2)i.

c = 2 - 3i.

So, to rotate z around c by π/4, we can think of it as:

First, translate the system so that c is at the origin: subtract c from z. Then rotate by π/4, then translate back by adding c.

So, as before.

Alternatively, using complex number operations.

Another way to think about rotation is using complex multiplication. If you have a complex number a, and you want to rotate it around the origin by θ, it's a * e^{iθ}. But when rotating around another point, you have to adjust.

But I think the method I used is correct.

Alternatively, maybe I can use the formula for rotation in complex plane.

Alternatively, think of the rotation as a combination of translation, rotation, and translation back.

But I think I have verified it multiple times.

Alternatively, maybe I can check with specific numbers. Let me compute z - c again:

z = 2 + √2 - (3 + 3√2)i

c = 2 - 3i

So z - c = (2 + √2 - 2) + [ - (3 + 3√2)i - (-3i) ] = √2 + [ -3 - 3√2 + 3 ]i = √2 - 3√2 i. Correct.

So that's √2(1 - 3i).

Multiply by e^{iπ/4} which is cos(π/4) + i sin(π/4) = √2/2 + i√2/2.

So multiplying √2(1 - 3i) by √2/2 + i√2/2.

Let me compute that:

First, √2*(√2/2) = (2)/2 = 1.

Then, √2*(i√2/2) = i*(2)/2 = i.

Then, -3i*(√2/2) = -3i√2/2.

Then, -3i*(i√2/2) = -3i²√2/2 = -3*(-1)√2/2 = 3√2/2.

So adding all these terms:

1 + i - 3i√2/2 + 3√2/2.

Wait, hold on, this seems different from before. Wait, maybe I did something wrong here. Wait, no, actually, I think I need to multiply (1 - 3i) by (√2/2 + i√2/2). Wait, no, the original expression is √2*(1 - 3i) multiplied by (√2/2 + i√2/2). Which is equal to √2*(√2/2 + i√2/2)*(1 - 3i).

Alternatively, maybe I should compute (1 - 3i)*(√2/2 + i√2/2) first, then multiply by √2? Wait, no, the √2 is already factored out. Wait, no, the original expression is √2*(1 - 3i) multiplied by (√2/2 + i√2/2). Let me compute that:

First, multiply √2*(√2/2 + i√2/2) = √2*(√2/2)(1 + i) = (2/2)(1 + i) = 1*(1 + i) = 1 + i. Then multiply by (1 - 3i). Wait, no, that's not correct. Wait, no, the original expression is √2*(1 - 3i) multiplied by (√2/2 + i√2/2). Which is equal to [√2*(√2/2 + i√2/2)]*(1 - 3i). Which is [ (2/2 + i*2/2 ) ]*(1 - 3i) = [1 + i]*(1 - 3i). Which is exactly what I did before. Which gives 4 - 2i. So that's correct.

So that approach also gives 4 - 2i. Then adding c gives 6 - 5i.

Therefore, I think my answer is correct.

But just to be 100% sure, let me compute the entire thing using decimal approximations. Let me compute numerical values.

First, compute z - c:

z = 2 + √2 - (3 + 3√2)i ≈ 2 + 1.4142 - (3 + 4.2426)i ≈ 3.4142 - 7.2426i

c = 2 - 3i

So z - c ≈ 3.4142 - 2 - (7.2426 - (-3))i? Wait no:

Wait z - c is (real part of z - real part of c) + (imaginary part of z - imaginary part of c)i.

So real part: 2 + √2 - 2 = √2 ≈ 1.4142

Imaginary part: -(3 + 3√2) - (-3) = -3 - 3√2 + 3 = -3√2 ≈ -4.2426

So z - c ≈ 1.4142 - 4.2426i

Now, multiply by e^{iπ/4} ≈ cos(π/4) + i sin(π/4) ≈ 0.7071 + 0.7071i

So multiplying (1.4142 - 4.2426i) * (0.7071 + 0.7071i)

Let me compute this:

First, multiply 1.4142 * 0.7071 ≈ 1.4142 * 0.7071 ≈ 1.0 (since √2 * √2/2 = 1)

Then, 1.4142 * 0.7071i ≈ 1.0i

Then, -4.2426i * 0.7071 ≈ -4.2426 * 0.7071 i ≈ -3.0i (since 4.2426 ≈ 3√2 ≈ 4.2426, and 3√2 * √2/2 = 3*2/2 = 3)

Then, -4.2426i * 0.7071i ≈ -4.2426 * 0.7071 * i² ≈ -3.0 * (-1) = 3.0

So adding all terms:

1.0 + 1.0i - 3.0i + 3.0 = (1.0 + 3.0) + (1.0 - 3.0)i = 4.0 - 2.0i.

Which matches the previous result. Then adding c = 2 - 3i:

4.0 - 2.0i + 2 - 3i = 6.0 - 5.0i.

So numerically, it's 6 - 5i. Therefore, this confirms the answer.

Therefore, I think the answer is 6 - 5i.

But just to check once more, let me think if there's any chance I messed up the direction of rotation. Since rotating counter-clockwise by π/4, which is the standard direction. If I had rotated clockwise, it would be e^{-iπ/4}, but the problem says counter-clockwise, so it's correct.

Alternatively, if I had used a different formula, like rotating around c, maybe I need to use a different approach? Let me think.

Another way to think about rotation: if you have a point P, and you want to rotate it around point Q by angle θ, then the formula is:

P' = Q + (P - Q) * e^{iθ}

Which is exactly what I used. So that's correct.

Therefore, I think my answer is correct.

**Final Answer**
\boxed{6 - 5i}

To find the complex number $ w $, which is the result of rotating $ z $ around $ c $ by $ \frac{\pi}{4} $ radians **counter-clockwise**, we use the standard formula for rotating a complex number around another point:

$$
w = (z - c) \cdot e^{i\theta} + c
$$

---

### Step 1: Identify the given values

- $ z = 2 + \sqrt{2} - (3 + 3\sqrt{2})i $
- $ c = 2 - 3i $
- Rotation angle $ \theta = \frac{\pi}{4} $

---

### Step 2: Compute $ z - c $

We subtract $ c $ from $ z $:

$$
z - c = \left(2 + \sqrt{2} - 2\right) + \left[-(3 + 3\sqrt{2})i - (-3i)\right]
$$

$$
= \sqrt{2} + \left[-3 - 3\sqrt{2} + 3\right]i = \sqrt{2} - 3\sqrt{2}i
$$

Factor out $ \sqrt{2} $:

$$
z - c = \sqrt{2}(1 - 3i)
$$

---

### Step 3: Multiply by $ e^{i\pi/4} $

We use the identity:

$$
e^{i\pi/4} = \cos\left(\frac{\pi}{4}\right) + i\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}
$$

So:

$$
(z - c) \cdot e^{i\pi/4} = \sqrt{2}(1 - 3i) \cdot \left(\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}\right)
$$

$$
= \sqrt{2} \cdot \left(1 - 3i\right) \cdot \left(\frac{\sqrt{2}}{2}(1 + i)\right)
$$

$$
= \left(1 - 3i\right) \cdot (1 + i)
$$

Now expand:

$$
(1 - 3i)(1 + i) = 1 + i - 3i - 3i^2 = 1 - 2i + 3 = 4 - 2i
$$

---

### Step 4: Add back $ c $

$$
w = (z - c) \cdot e^{i\pi/4} + c = (4 - 2i) + (2 - 3i) = 6 - 5i
$$

---

### Final Answer

$$
\boxed{6 - 5i}
$$

[Assistant]
Explanation:
Okay, so I need to find the complex number w, which is the result of rotating z around c by π/4 radians counter-clockwise. Let me recall how to perform such rotations in the complex plane.

First, I remember that rotating a complex number around another point involves translating the complex plane so that the center of rotation becomes the origin, performing the rotation, and then translating back. So, the general formula for rotating a point z around a point c by an angle θ is:

w = (z - c) * e^{iθ} + c

Yes, that seems right. Let me verify. If I subtract c from z, that moves the center of rotation to the origin. Then multiplying by e^{iθ} rotates it by θ, and then adding c brings it back to the original position. So, that formula should work.

Given that, let me write down the given values.

z is given as 2 + √2 - (3 + 3√2)i. Let me write that in standard form. The real part is 2 + √2, and the imaginary part is -(3 + 3√2). So, z = (2 + √2) - (3 + 3√2)i.

c is given as 2 - 3i. So, c = 2 - 3i.

The angle θ is π/4 radians, which is 45 degrees. The rotation is counter-clockwise, so the rotation factor is e^{iπ/4}.

So, applying the formula:

w = (z - c) * e^{iπ/4} + c

First, I need to compute z - c. Let me compute that.

z - c = [ (2 + √2) - (3 + 3√2)i ] - [2 - 3i]

Let me subtract the real parts and the imaginary parts separately.

Real part: (2 + √2) - 2 = √2

Imaginary part: [ - (3 + 3√2)i ] - (-3i) = -3 - 3√2 + 3 = -3√2 i

Wait, let me check that again. The imaginary part of z is -(3 + 3√2)i, and the imaginary part of c is -3i. So when subtracting c, which is subtracting 2 - 3i, so subtracting the real part and subtracting the imaginary part. Wait, actually, when you subtract c from z, it's z - c = (real part of z - real part of c) + (imaginary part of z - imaginary part of c)i.

So, real part: (2 + √2) - 2 = √2. Correct.

Imaginary part: [ - (3 + 3√2) ] - (-3) = -3 - 3√2 + 3 = -3√2. So the imaginary part is -3√2. Therefore, z - c = √2 - 3√2 i.

So z - c = √2(1 - 3i). Let me write that as √2*(1 - 3i).

Now, I need to multiply this by e^{iπ/4}. Let me recall that e^{iπ/4} is cos(π/4) + i sin(π/4) = (√2/2) + i(√2/2).

Alternatively, I can write e^{iπ/4} as (1 + i)/√2. Let me confirm:

(1 + i)/√2 has magnitude √( (1/√2)^2 + (1/√2)^2 ) = √(1/2 + 1/2) = √1 = 1, and angle arctan( (1/√2)/(1/√2) ) = arctan(1) = π/4. Yes, correct. So e^{iπ/4} = (1 + i)/√2.

Therefore, multiplying (z - c) by e^{iπ/4} is equivalent to multiplying √2*(1 - 3i) by (1 + i)/√2. The √2 cancels with 1/√2, so we get (1 - 3i)(1 + i).

Let me compute that:

(1 - 3i)(1 + i) = 1*(1) + 1*(i) - 3i*(1) - 3i*(i) = 1 + i - 3i - 3i².

Simplify:

1 + i - 3i - 3(-1) = 1 - 2i + 3 = 4 - 2i.

So, the product (z - c)*e^{iπ/4} is 4 - 2i. Then, adding back c, which is 2 - 3i, gives:

w = (4 - 2i) + (2 - 3i) = (4 + 2) + (-2i - 3i) = 6 - 5i.

Wait, so is that the answer? Let me check my steps again to make sure I didn't make a mistake.

First, z - c:

z = 2 + √2 - (3 + 3√2)i

c = 2 - 3i

So z - c = (2 + √2 - 2) + [ - (3 + 3√2)i - (-3i) ] = √2 + [ -3 - 3√2 + 3 ]i = √2 - 3√2 i. Correct. So that's √2(1 - 3i). Then multiplying by e^{iπ/4} which is (1 + i)/√2. So √2*(1 - 3i)*(1 + i)/√2 = (1 - 3i)(1 + i). Then expanding that:

1*1 + 1*i - 3i*1 - 3i*i = 1 + i - 3i - 3i². Since i² = -1, so -3i² = 3. Therefore, 1 + 3 + i - 3i = 4 - 2i. Correct. Then adding c, which is 2 - 3i. So 4 - 2i + 2 - 3i = 6 - 5i.

Wait, but let me check if the rotation direction is correct. The problem says counter-clockwise, which is the standard direction for positive angles. Since we used e^{iπ/4}, which is a counter-clockwise rotation, that's correct. So seems okay.

But let me check with another approach. Maybe using rotation matrices?

Alternatively, since complex numbers can be represented as vectors, and rotation can be done with rotation matrices. Let me try that approach to verify.

First, z - c is the vector from c to z. So, if I consider z - c as a complex number, which we found to be √2 - 3√2 i. Let me write that as a vector ( √2, -3√2 ). Then, rotating this vector by π/4 counter-clockwise. The rotation matrix for π/4 is:

[ cos(π/4)  -sin(π/4) ]
[ sin(π/4)   cos(π/4) ]

Which is:

[ √2/2   -√2/2 ]
[ √2/2    √2/2 ]

So multiplying this matrix by the vector ( √2, -3√2 ):

First component: (√2/2)(√2) + (-√2/2)(-3√2)

Second component: (√2/2)(√2) + (√2/2)(-3√2)

Let me compute first component:

(√2/2)(√2) = (2)/2 = 1

(-√2/2)(-3√2) = (3*(√2)^2)/2 = 3*2/2 = 3

So total first component: 1 + 3 = 4

Second component:

(√2/2)(√2) = 1

(√2/2)(-3√2) = -3*(√2)^2 / 2 = -3*2 / 2 = -3

So total second component: 1 - 3 = -2

Therefore, the rotated vector is (4, -2), which corresponds to the complex number 4 - 2i. Then adding back c, which is (2, -3), so adding 2 - 3i gives (4 + 2, -2 + (-3)) = (6, -5), which is 6 - 5i. So same result. Therefore, seems correct.

But let me check once more with another method. Maybe converting to polar form?

But z - c is √2 - 3√2 i. Let me write that as √2(1 - 3i). Let me find its magnitude and angle.

Magnitude: |√2(1 - 3i)| = √2 * |1 - 3i| = √2 * √(1^2 + (-3)^2) = √2 * √(1 + 9) = √2 * √10 = √20 = 2√5.

Angle: arctangent of (-3)/1 = arctan(-3). Since the complex number is in the fourth quadrant (positive real, negative imaginary). So angle θ = -arctan(3) or equivalently 2π - arctan(3).

But when we rotate by π/4, the new angle will be θ + π/4. Then, converting back to rectangular form. However, this might be more complicated, but let me try.

Alternatively, since we already have the result via two methods, maybe it's safe. But let me check if the answer is 6 - 5i. Wait, but let me check with the Asymptote code given. The Asymptote code defines Z as (2 + sqrt(2), -3 - 3*sqrt(2)), which is the coordinates of z. Let me check if that matches with z.

Original z is 2 + sqrt(2) - (3 + 3 sqrt(2))i. So in coordinates, that would be (real part, imaginary part) = (2 + sqrt(2), - (3 + 3 sqrt(2))). Which is (2 + sqrt(2), -3 - 3 sqrt(2)). So yes, that's what the Asymptote code says. Then c is (2, -3). Then W is rotate(45, C)*(Z). So the Asymptote code is rotating Z around C by 45 degrees. So according to the Asymptote code, the answer should be the coordinates of W. Let me compute coordinates of W.

If my answer is 6 - 5i, then coordinates are (6, -5). Let me check if that's consistent with the rotation.

Alternatively, maybe I can compute coordinates step by step. Let me do that.

Original point Z is (2 + √2, -3 - 3√2). Center C is (2, -3). So vector from C to Z is ( (2 + √2 - 2), (-3 - 3√2 - (-3)) ) = (√2, -3√2). Which is same as before. Rotating this vector by π/4 gives (4, -2) as before. Then adding back C's coordinates (2, -3) gives (4 + 2, -2 + (-3)) = (6, -5). So yes, coordinates (6, -5), which is 6 - 5i. So that seems correct.

But let me check if the Asymptote code's drawing is consistent. The Asymptote code draws Z--C--W. So from Z to C to W. So if Z is (2 + sqrt(2), -3 - 3 sqrt(2)), C is (2, -3), and W is (6, -5). Let me check the distance from Z to C and from C to W. Wait, but since it's a rotation, the distance from Z to C should be same as from W to C. Let me check.

Distance from Z to C: sqrt( (√2)^2 + (-3√2)^2 ) = sqrt( 2 + 9*2 ) = sqrt(2 + 18) = sqrt(20) = 2√5.

Distance from W to C: sqrt( (6 - 2)^2 + (-5 - (-3))^2 ) = sqrt( (4)^2 + (-2)^2 ) = sqrt(16 + 4) = sqrt(20) = 2√5. So same distance. Good. Also, the angle between ZC and WC should be π/4. Let me check the angle.

But maybe that's too involved. Alternatively, since all methods give me 6 - 5i, and the Asymptote code seems to confirm the coordinates, I think that is correct.

But just to be thorough, let me check once again the multiplication step.

(z - c) = √2 - 3√2 i. Multiply by e^{iπ/4} = (1 + i)/√2.

So:

(√2 - 3√2 i) * (1 + i)/√2 = [√2(1 - 3i)] * (1 + i)/√2 = (1 - 3i)(1 + i) = 1*(1) + 1*(i) - 3i*(1) - 3i*(i) = 1 + i - 3i - 3i² = 1 - 2i + 3 = 4 - 2i. Correct. Then adding c = 2 - 3i gives 4 - 2i + 2 - 3i = 6 - 5i.

Yes, that's correct. Therefore, I think the answer is 6 - 5i.

But just to make sure, let me try another approach. Let me represent all numbers as complex numbers and perform the rotation step by step.

Let me write z = 2 + √2 - (3 + 3√2)i. Let me write that as z = (2 + √2) + (-3 - 3√2)i.

c = 2 - 3i.

So, to rotate z around c by π/4, we can think of it as:

First, translate the system so that c is at the origin: subtract c from z. Then rotate by π/4, then translate back by adding c.

So, as before.

Alternatively, using complex number operations.

Another way to think about rotation is using complex multiplication. If you have a complex number a, and you want to rotate it around the origin by θ, it's a * e^{iθ}. But when rotating around another point, you have to adjust.

But I think the method I used is correct.

Alternatively, maybe I can use the formula for rotation in complex plane.

Alternatively, think of the rotation as a combination of translation, rotation, and translation back.

But I think I have verified it multiple times.

Alternatively, maybe I can check with specific numbers. Let me compute z - c again:

z = 2 + √2 - (3 + 3√2)i

c = 2 - 3i

So z - c = (2 + √2 - 2) + [ - (3 + 3√2)i - (-3i) ] = √2 + [ -3 - 3√2 + 3 ]i = √2 - 3√2 i. Correct.

So that's √2(1 - 3i).

Multiply by e^{iπ/4} which is cos(π/4) + i sin(π/4) = √2/2 + i√2/2.

So multiplying √2(1 - 3i) by √2/2 + i√2/2.

Let me compute that:

First, √2*(√2/2) = (2)/2 = 1.

Then, √2*(i√2/2) = i*(2)/2 = i.

Then, -3i*(√2/2) = -3i√2/2.

Then, -3i*(i√2/2) = -3i²√2/2 = -3*(-1)√2/2 = 3√2/2.

So adding all these terms:

1 + i - 3i√2/2 + 3√2/2.

Wait, hold on, this seems different from before. Wait, maybe I did something wrong here. Wait, no, actually, I think I need to multiply (1 - 3i) by (√2/2 + i√2/2). Wait, no, the original expression is √2*(1 - 3i) multiplied by (√2/2 + i√2/2). Which is equal to √2*(√2/2 + i√2/2)*(1 - 3i).

Alternatively, maybe I should compute (1 - 3i)*(√2/2 + i√2/2) first, then multiply by √2? Wait, no, the √2 is already factored out. Wait, no, the original expression is √2*(1 - 3i) multiplied by (√2/2 + i√2/2). Let me compute that:

First, multiply √2*(√2/2 + i√2/2) = √2*(√2/2)(1 + i) = (2/2)(1 + i) = 1*(1 + i) = 1 + i. Then multiply by (1 - 3i). Wait, no, that's not correct. Wait, no, the original expression is √2*(1 - 3i) multiplied by (√2/2 + i√2/2). Which is equal to [√2*(√2/2 + i√2/2)]*(1 - 3i). Which is [ (2/2 + i*2/2 ) ]*(1 - 3i) = [1 + i]*(1 - 3i). Which is exactly what I did before. Which gives 4 - 2i. So that's correct.

So that approach also gives 4 - 2i. Then adding c gives 6 - 5i.

Therefore, I think my answer is correct.

But just to be 100% sure, let me compute the entire thing using decimal approximations. Let me compute numerical values.

First, compute z - c:

z = 2 + √2 - (3 + 3√2)i ≈ 2 + 1.4142 - (3 + 4.2426)i ≈ 3.4142 - 7.2426i

c = 2 - 3i

So z - c ≈ 3.4142 - 2 - (7.2426 - (-3))i? Wait no:

Wait z - c is (real part of z - real part of c) + (imaginary part of z - imaginary part of c)i.

So real part: 2 + √2 - 2 = √2 ≈ 1.4142

Imaginary part: -(3 + 3√2) - (-3) = -3 - 3√2 + 3 = -3√2 ≈ -4.2426

So z - c ≈ 1.4142 - 4.2426i

Now, multiply by e^{iπ/4} ≈ cos(π/4) + i sin(π/4) ≈ 0.7071 + 0.7071i

So multiplying (1.4142 - 4.2426i) * (0.7071 + 0.7071i)

Let me compute this:

First, multiply 1.4142 * 0.7071 ≈ 1.4142 * 0.7071 ≈ 1.0 (since √2 * √2/2 = 1)

Then, 1.4142 * 0.7071i ≈ 1.0i

Then, -4.2426i * 0.7071 ≈ -4.2426 * 0.7071 i ≈ -3.0i (since 4.2426 ≈ 3√2 ≈ 4.2426, and 3√2 * √2/2 = 3*2/2 = 3)

Then, -4.2426i * 0.7071i ≈ -4.2426 * 0.7071 * i² ≈ -3.0 * (-1) = 3.0

So adding all terms:

1.0 + 1.0i - 3.0i + 3.0 = (1.0 + 3.0) + (1.0 - 3.0)i = 4.0 - 2.0i.

Which matches the previous result. Then adding c = 2 - 3i:

4.0 - 2.0i + 2 - 3i = 6.0 - 5.0i.

So numerically, it's 6 - 5i. Therefore, this confirms the answer.

Therefore, I think the answer is 6 - 5i.

But just to check once more, let me think if there's any chance I messed up the direction of rotation. Since rotating counter-clockwise by π/4, which is the standard direction. If I had rotated clockwise, it would be e^{-iπ/4}, but the problem says counter-clockwise, so it's correct.

Alternatively, if I had used a different formula, like rotating around c, maybe I need to use a different approach? Let me think.

Another way to think about rotation: if you have a point P, and you want to rotate it around point Q by angle θ, then the formula is:

P' = Q + (P - Q) * e^{iθ}

Which is exactly what I used. So that's correct.

Therefore, I think my answer is correct.

**Final Answer**
\boxed{6 - 5i}

To find the complex number $ w $, which is the result of rotating $ z $ around $ c $ by $ \frac{\pi}{4} $ radians **counter-clockwise**, we use the standard formula for rotating a complex number around another point:

$$
w = (z - c) \cdot e^{i\theta} + c
$$

---

### Step 1: Identify the given values

- $ z = 2 + \sqrt{2} - (3 + 3\sqrt{2})i $
- $ c = 2 - 3i $
- Rotation angle $ \theta = \frac{\pi}{4} $

---

### Step 2: Compute $ z - c $

We subtract $ c $ from $ z $:

$$
z - c = \left(2 + \sqrt{2} - 2\right) + \left[-(3 + 3\sqrt{2})i - (-3i)\right]
$$

$$
= \sqrt{2} + \left[-3 - 3\sqrt{2} + 3\right]i = \sqrt{2} - 3\sqrt{2}i
$$

Factor out $ \sqrt{2} $:

$$
z - c = \sqrt{2}(1 - 3i)
$$

---

### Step 3: Multiply by $ e^{i\pi/4} $

We use the identity:

$$
e^{i\pi/4} = \cos\left(\frac{\pi}{4}\right) + i\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}
$$

So:

$$
(z - c) \cdot e^{i\pi/4} = \sqrt{2}(1 - 3i) \cdot \left(\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}\right)
$$

$$
= \sqrt{2} \cdot \left(1 - 3i\right) \cdot \left(\frac{\sqrt{2}}{2}(1 + i)\right)
$$

$$
= \left(1 - 3i\right) \cdot (1 + i)
$$

Now expand:

$$
(1 - 3i)(1 + i) = 1 + i - 3i - 3i^2 = 1 - 2i + 3 = 4 - 2i
$$

---

### Step 4: Add back $ c $

$$
w = (z - c) \cdot e^{i\pi/4} + c = (4 - 2i) + (2 - 3i) = 6 - 5i
$$

---

### Final Answer

$$
\boxed{6 - 5i}
$$

Solution:
Rotating by $\frac{\pi}{4}$ counter-clockwise corresponds to the complex number
\[e^{\pi i/4} = \frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}}.\]Hence,
\[w - c = \left( \frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}} \right) (z - c),\]so
\begin{align*}
w &= \left( \frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}} \right) (z - c) + c \\
&= \left( \frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}} \right) (\sqrt{2} - 3i \sqrt{2}) + 2 - 3i \\
&= (4 - 2i) + 2 - 3i \\
&= \boxed{6 - 5i}.
\end{align*}


Explanation:
Okay, let's try to tackle this problem. So, Denali and Nate are dog walkers, right? They get paid based on the number of dogs they walk. Denali has 16 dogs, and Nate has 12. Now, there's this new policy where they can be assigned or unassigned new dogs in groups of x dogs. The problem says that the ratio of Denali's pay to Nate's pay would be the same in two different scenarios:

1. If Denali starts walking 4x more dogs and Nate stays at 12 dogs.
2. If x of Nate's dogs are reassigned to Denali.

We need to find x, given that x is not zero. Alright, let me break this down step by step.

First, let me make sure I understand the problem correctly. The original number of dogs each is responsible for is 16 for Denali and 12 for Nate. Then, there are two different scenarios where their number of dogs changes, and in both scenarios, the ratio of their pay remains the same. The pay ratio depends on the number of dogs they walk, so I assume that the pay is directly proportional to the number of dogs. Therefore, the ratio of their pay is the same as the ratio of their number of dogs.

So, originally, the ratio of Denali's pay to Nate's pay is 16:12, which simplifies to 4:3. But the problem says that this ratio remains the same in two different scenarios. Let me check if that's the case.

Wait, actually, the problem says that the ratio would be the same if either Denali gets 4x more dogs or if x of Nate's dogs are reassigned to Denali. So, the original ratio is 16:12, but in the two scenarios, the ratio is the same as the original? Or is it that the ratio is the same between the two scenarios? Wait, the wording is: "The ratio of Denali's pay to Nate's pay would be the same if Denali started walking 4x more dogs and Nate stayed at 12 dogs or if x of Nate's dogs were reassigned to Denali."

So, the ratio is the same in both cases. So, in both cases, the ratio of Denali's pay to Nate's pay is equal. Therefore, we need to set up equations for both scenarios and set them equal to each other.

Let me formalize this.

Original scenario: Denali has 16 dogs, Nate has 12 dogs. So, ratio is 16/12 = 4/3.

First scenario: Denali starts walking 4x more dogs. So, Denali's dogs become 16 + 4x, Nate's dogs remain 12. Therefore, the ratio is (16 + 4x)/12.

Second scenario: x of Nate's dogs are reassigned to Denali. So, Denali's dogs become 16 + x, and Nate's dogs become 12 - x. Therefore, the ratio is (16 + x)/(12 - x).

Since the ratio is the same in both scenarios, we can set these two expressions equal to each other:

(16 + 4x)/12 = (16 + x)/(12 - x)

Now, we need to solve for x. Let me write that equation again:

(16 + 4x)/12 = (16 + x)/(12 - x)

Okay, cross-multiplying to eliminate denominators:

(16 + 4x)(12 - x) = 12(16 + x)

Let me expand both sides.

First, left side: (16 + 4x)(12 - x)

Multiply term by term:

16*12 + 16*(-x) + 4x*12 + 4x*(-x)

Which is 192 - 16x + 48x - 4x²

Combine like terms:

192 + ( -16x + 48x ) - 4x² = 192 + 32x - 4x²

Right side: 12*(16 + x) = 12*16 + 12x = 192 + 12x

So, the equation becomes:

192 + 32x - 4x² = 192 + 12x

Let me subtract 192 from both sides:

32x - 4x² = 12x

Subtract 12x from both sides:

20x - 4x² = 0

Let me rearrange:

-4x² + 20x = 0

Multiply both sides by -1 to make it more standard:

4x² - 20x = 0

Factor out 4x:

4x(x - 5) = 0

So, the solutions are 4x = 0 => x = 0 or x - 5 = 0 => x = 5.

But the problem states that x ≠ 0, so x = 5.

Wait, that seems straightforward. Let me check if this answer makes sense.

Let me verify by plugging x = 5 into both scenarios.

First scenario: Denali has 16 + 4x = 16 + 20 = 36 dogs. Nate has 12. So, ratio is 36/12 = 3/1.

Second scenario: Denali has 16 + x = 16 + 5 = 21 dogs. Nate has 12 - x = 12 - 5 = 7 dogs. So, ratio is 21/7 = 3/1.

So, both ratios are 3:1, which is the same. Therefore, x = 5 is correct.

But let me check if there's another way to interpret the problem. The problem mentions that they are assigned or unassigned new dogs in groups of x dogs. So, does that mean that the new dogs are added in groups of x? But in the first scenario, Denali is getting 4x more dogs. So, if they are assigned in groups of x, then 4x would be 4 groups of x dogs. Similarly, in the second scenario, x of Nate's dogs are reassigned. So, maybe that's just moving x dogs from Nate to Denali.

But the problem says "assigned or unassigned new dogs in groups of x dogs." So, maybe the new dogs are being added in groups of x. However, in the first scenario, Denali is getting 4x more dogs. So, that would be 4 groups of x dogs. So, that seems okay.

But maybe there's a different interpretation? For example, maybe the total number of dogs they have is being adjusted by adding or removing groups of x dogs. But the problem says "they will be assigned or unassigned new dogs in groups of x dogs." So, perhaps when they are assigned new dogs, it's in groups of x, and when they are unassigned, it's also in groups of x.

But in the first scenario, Denali is starting to walk 4x more dogs. So, that would be adding 4x dogs. So, if they are assigned in groups of x, then adding 4x would be adding 4 groups. Similarly, in the second scenario, x of Nate's dogs are reassigned to Denali. So, that would be removing x dogs from Nate and adding them to Denali. So, that's a reassignment of x dogs.

But maybe the problem is that the original number of dogs is 16 and 12, and then they can be assigned or unassigned in groups of x. So, maybe the total number of dogs is changing? But the problem says "the ratio of Denali's pay to Nate's pay would be the same if Denali started walking 4x more dogs and Nate stayed at 12 dogs or if x of Nate's dogs were reassigned to Denali."

So, the first case is Denali has 16 + 4x, Nate has 12. The second case is Denali has 16 + x, Nate has 12 - x. So, the equations are as I set up before.

Therefore, solving gives x = 5. Which checks out.

But let me check if there's another possible interpretation. For example, maybe the ratio is supposed to be the same as the original ratio? Wait, the original ratio is 16/12 = 4/3. But in the first scenario, with x = 5, the ratio is 36/12 = 3/1, which is different from 4/3. However, the problem says "the ratio of Denali's pay to Nate's pay would be the same if Denali started walking 4x more dogs and Nate stayed at 12 dogs or if x of Nate's dogs were reassigned to Denali." So, the ratio in both scenarios is the same, but not necessarily the same as the original ratio.

So, in this case, the original ratio is 4/3, but in the two scenarios, the ratio is 3/1. So, the problem is not saying that the ratio is the same as the original ratio, but that the ratio is the same in both scenarios. Therefore, my initial approach is correct.

Alternatively, maybe the problem is saying that the ratio is the same as the original ratio? Let me check. If that were the case, then we would have:

Original ratio: 16/12 = 4/3.

First scenario: (16 + 4x)/12 = 4/3.

Second scenario: (16 + x)/(12 - x) = 4/3.

But solving these equations would give different x. Let me check if that's possible. However, the problem says "the ratio of Denali's pay to Nate's pay would be the same if Denali started walking 4x more dogs and Nate stayed at 12 dogs or if x of Nate's dogs were reassigned to Denali." So, the ratio in the first case is equal to the ratio in the second case, not necessarily equal to the original ratio. Therefore, my original interpretation is correct.

But just to be thorough, let me check if the problem might have meant that the ratio is the same as the original ratio. If that were the case, then solving (16 + 4x)/12 = 4/3 would give:

(16 + 4x)/12 = 4/3

Multiply both sides by 12:

16 + 4x = 16

Therefore, 4x = 0 => x = 0. Which is not allowed. Similarly, for the second scenario:

(16 + x)/(12 - x) = 4/3

Cross-multiplying:

3(16 + x) = 4(12 - x)

48 + 3x = 48 - 4x

3x + 4x = 48 - 48

7x = 0 => x = 0. Again, same result. Therefore, if the problem had meant that the ratio is the same as the original ratio, then x would be zero, which is invalid. Therefore, the problem must mean that the ratio in the two scenarios is the same, not necessarily the original ratio. Therefore, my original solution of x = 5 is correct.

Another check: Let me think if there's any constraints on x. For example, in the second scenario, Nate has 12 - x dogs. Since you can't have negative dogs, 12 - x must be greater than 0. Therefore, x < 12. Since x = 5, that's okay. Also, in the first scenario, Denali has 16 + 4x dogs. Since x is positive, that's fine. So, x = 5 is valid.

Therefore, I think the answer is 5.

But let me just go through the algebra again to make sure I didn't make any mistakes.

Original equation:

(16 + 4x)/12 = (16 + x)/(12 - x)

Cross-multiplying:

(16 + 4x)(12 - x) = 12(16 + x)

Expanding left side:

16*12 - 16x + 4x*12 - 4x^2 = 192 -16x + 48x -4x² = 192 + 32x -4x²

Right side: 12*16 + 12x = 192 + 12x

Set equal:

192 + 32x -4x² = 192 + 12x

Subtract 192:

32x -4x² = 12x

Subtract 12x:

20x -4x² = 0

Factor:

-4x² +20x = 0 => 4x² -20x = 0 => 4x(x -5) = 0 => x=0 or x=5. Correct.

Yes, that's correct. So, x=5 is the answer.

Therefore, I think the answer is 5.

**Final Answer**
The value of $ x $ is \boxed{5}.

To solve the problem, we are given that Denali and Nate are responsible for walking a certain number of dogs, and under a new policy, they can be assigned or reassigned dogs in groups of $ x $ dogs. The key condition is that the **ratio of Denali's pay to Nate's pay** remains the same in two different scenarios:

1. **Scenario 1**: Denali walks $ 4x $ more dogs, while Nate remains at 12 dogs.
2. **Scenario 2**: $ x $ of Nate's dogs are reassigned to Denali.

We are to find the value of $ x $, given that $ x \neq 0 $.

---

### Step 1: Express the Pay Ratios

Assuming pay is directly proportional to the number of dogs walked, the ratio of their pay is the same as the ratio of the number of dogs they walk.

- **Original scenario**: Denali walks 16 dogs, Nate walks 12 dogs → ratio = $ \frac{16}{12} = \frac{4}{3} $
- **Scenario 1**: Denali walks $ 16 + 4x $, Nate walks 12 → ratio = $ \frac{16 + 4x}{12} $
- **Scenario 2**: Denali walks $ 16 + x $, Nate walks $ 12 - x $ → ratio = $ \frac{16 + x}{12 - x} $

We are told that the ratio is the same in both scenarios, so we set the two expressions equal:

$$
\frac{16 + 4x}{12} = \frac{16 + x}{12 - x}
$$

---

### Step 2: Solve the Equation

Cross-multiply to eliminate the denominators:

$$
(16 + 4x)(12 - x) = 12(16 + x)
$$

Expand both sides:

**Left side:**
$$
(16 + 4x)(12 - x) = 16 \cdot 12 - 16x + 4x \cdot 12 - 4x^2 = 192 - 16x + 48x - 4x^2 = 192 + 32x - 4x^2
$$

**Right side:**
$$
12(16 + x) = 192 + 12x
$$

Now equate both sides:

$$
192 + 32x - 4x^2 = 192 + 12x
$$

Subtract 192 from both sides:

$$
32x - 4x^2 = 12x
$$

Subtract $ 12x $ from both sides:

$$
20x - 4x^2 = 0
$$

Factor:

$$
4x(5 - x) = 0
$$

So, the solutions are:

$$
x = 0 \quad \text{or} \quad x = 5
$$

Since $ x \neq 0 $, the only valid solution is:

$$
x = 5
$$

---

### Step 3: Verify the Solution

Let’s verify that the ratio is the same in both scenarios when $ x = 5 $:

- **Scenario 1**: Denali walks $ 16 + 4(5) = 36 $, Nate walks 12 → ratio = $ \frac{36}{12} = 3 $
- **Scenario 2**: Denali walks $ 16 + 5 = 21 $, Nate walks $ 12 - 5 = 7 $ → ratio = $ \frac{21}{7} = 3 $

Both ratios are equal, confirming that the solution is correct.

---

### Final Answer

$$
\boxed{5}
$$

[Assistant]
Explanation:
Okay, let's try to tackle this problem. So, Denali and Nate are dog walkers, right? They get paid based on the number of dogs they walk. Denali has 16 dogs, and Nate has 12. Now, there's this new policy where they can be assigned or unassigned new dogs in groups of x dogs. The problem says that the ratio of Denali's pay to Nate's pay would be the same in two different scenarios:

1. If Denali starts walking 4x more dogs and Nate stays at 12 dogs.
2. If x of Nate's dogs are reassigned to Denali.

We need to find x, given that x is not zero. Alright, let me break this down step by step.

First, let me make sure I understand the problem correctly. The original number of dogs each is responsible for is 16 for Denali and 12 for Nate. Then, there are two different scenarios where their number of dogs changes, and in both scenarios, the ratio of their pay remains the same. The pay ratio depends on the number of dogs they walk, so I assume that the pay is directly proportional to the number of dogs. Therefore, the ratio of their pay is the same as the ratio of their number of dogs.

So, originally, the ratio of Denali's pay to Nate's pay is 16:12, which simplifies to 4:3. But the problem says that this ratio remains the same in two different scenarios. Let me check if that's the case.

Wait, actually, the problem says that the ratio would be the same if either Denali gets 4x more dogs or if x of Nate's dogs are reassigned to Denali. So, the original ratio is 16:12, but in the two scenarios, the ratio is the same as the original? Or is it that the ratio is the same between the two scenarios? Wait, the wording is: "The ratio of Denali's pay to Nate's pay would be the same if Denali started walking 4x more dogs and Nate stayed at 12 dogs or if x of Nate's dogs were reassigned to Denali."

So, the ratio is the same in both cases. So, in both cases, the ratio of Denali's pay to Nate's pay is equal. Therefore, we need to set up equations for both scenarios and set them equal to each other.

Let me formalize this.

Original scenario: Denali has 16 dogs, Nate has 12 dogs. So, ratio is 16/12 = 4/3.

First scenario: Denali starts walking 4x more dogs. So, Denali's dogs become 16 + 4x, Nate's dogs remain 12. Therefore, the ratio is (16 + 4x)/12.

Second scenario: x of Nate's dogs are reassigned to Denali. So, Denali's dogs become 16 + x, and Nate's dogs become 12 - x. Therefore, the ratio is (16 + x)/(12 - x).

Since the ratio is the same in both scenarios, we can set these two expressions equal to each other:

(16 + 4x)/12 = (16 + x)/(12 - x)

Now, we need to solve for x. Let me write that equation again:

(16 + 4x)/12 = (16 + x)/(12 - x)

Okay, cross-multiplying to eliminate denominators:

(16 + 4x)(12 - x) = 12(16 + x)

Let me expand both sides.

First, left side: (16 + 4x)(12 - x)

Multiply term by term:

16*12 + 16*(-x) + 4x*12 + 4x*(-x)

Which is 192 - 16x + 48x - 4x²

Combine like terms:

192 + ( -16x + 48x ) - 4x² = 192 + 32x - 4x²

Right side: 12*(16 + x) = 12*16 + 12x = 192 + 12x

So, the equation becomes:

192 + 32x - 4x² = 192 + 12x

Let me subtract 192 from both sides:

32x - 4x² = 12x

Subtract 12x from both sides:

20x - 4x² = 0

Let me rearrange:

-4x² + 20x = 0

Multiply both sides by -1 to make it more standard:

4x² - 20x = 0

Factor out 4x:

4x(x - 5) = 0

So, the solutions are 4x = 0 => x = 0 or x - 5 = 0 => x = 5.

But the problem states that x ≠ 0, so x = 5.

Wait, that seems straightforward. Let me check if this answer makes sense.

Let me verify by plugging x = 5 into both scenarios.

First scenario: Denali has 16 + 4x = 16 + 20 = 36 dogs. Nate has 12. So, ratio is 36/12 = 3/1.

Second scenario: Denali has 16 + x = 16 + 5 = 21 dogs. Nate has 12 - x = 12 - 5 = 7 dogs. So, ratio is 21/7 = 3/1.

So, both ratios are 3:1, which is the same. Therefore, x = 5 is correct.

But let me check if there's another way to interpret the problem. The problem mentions that they are assigned or unassigned new dogs in groups of x dogs. So, does that mean that the new dogs are added in groups of x? But in the first scenario, Denali is getting 4x more dogs. So, if they are assigned in groups of x, then 4x would be 4 groups of x dogs. Similarly, in the second scenario, x of Nate's dogs are reassigned. So, maybe that's just moving x dogs from Nate to Denali.

But the problem says "assigned or unassigned new dogs in groups of x dogs." So, maybe the new dogs are being added in groups of x. However, in the first scenario, Denali is getting 4x more dogs. So, that would be 4 groups of x dogs. So, that seems okay.

But maybe there's a different interpretation? For example, maybe the total number of dogs they have is being adjusted by adding or removing groups of x dogs. But the problem says "they will be assigned or unassigned new dogs in groups of x dogs." So, perhaps when they are assigned new dogs, it's in groups of x, and when they are unassigned, it's also in groups of x.

But in the first scenario, Denali is starting to walk 4x more dogs. So, that would be adding 4x dogs. So, if they are assigned in groups of x, then adding 4x would be adding 4 groups. Similarly, in the second scenario, x of Nate's dogs are reassigned to Denali. So, that would be removing x dogs from Nate and adding them to Denali. So, that's a reassignment of x dogs.

But maybe the problem is that the original number of dogs is 16 and 12, and then they can be assigned or unassigned in groups of x. So, maybe the total number of dogs is changing? But the problem says "the ratio of Denali's pay to Nate's pay would be the same if Denali started walking 4x more dogs and Nate stayed at 12 dogs or if x of Nate's dogs were reassigned to Denali."

So, the first case is Denali has 16 + 4x, Nate has 12. The second case is Denali has 16 + x, Nate has 12 - x. So, the equations are as I set up before.

Therefore, solving gives x = 5. Which checks out.

But let me check if there's another possible interpretation. For example, maybe the ratio is supposed to be the same as the original ratio? Wait, the original ratio is 16/12 = 4/3. But in the first scenario, with x = 5, the ratio is 36/12 = 3/1, which is different from 4/3. However, the problem says "the ratio of Denali's pay to Nate's pay would be the same if Denali started walking 4x more dogs and Nate stayed at 12 dogs or if x of Nate's dogs were reassigned to Denali." So, the ratio in both scenarios is the same, but not necessarily the same as the original ratio.

So, in this case, the original ratio is 4/3, but in the two scenarios, the ratio is 3/1. So, the problem is not saying that the ratio is the same as the original ratio, but that the ratio is the same in both scenarios. Therefore, my initial approach is correct.

Alternatively, maybe the problem is saying that the ratio is the same as the original ratio? Let me check. If that were the case, then we would have:

Original ratio: 16/12 = 4/3.

First scenario: (16 + 4x)/12 = 4/3.

Second scenario: (16 + x)/(12 - x) = 4/3.

But solving these equations would give different x. Let me check if that's possible. However, the problem says "the ratio of Denali's pay to Nate's pay would be the same if Denali started walking 4x more dogs and Nate stayed at 12 dogs or if x of Nate's dogs were reassigned to Denali." So, the ratio in the first case is equal to the ratio in the second case, not necessarily equal to the original ratio. Therefore, my original interpretation is correct.

But just to be thorough, let me check if the problem might have meant that the ratio is the same as the original ratio. If that were the case, then solving (16 + 4x)/12 = 4/3 would give:

(16 + 4x)/12 = 4/3

Multiply both sides by 12:

16 + 4x = 16

Therefore, 4x = 0 => x = 0. Which is not allowed. Similarly, for the second scenario:

(16 + x)/(12 - x) = 4/3

Cross-multiplying:

3(16 + x) = 4(12 - x)

48 + 3x = 48 - 4x

3x + 4x = 48 - 48

7x = 0 => x = 0. Again, same result. Therefore, if the problem had meant that the ratio is the same as the original ratio, then x would be zero, which is invalid. Therefore, the problem must mean that the ratio in the two scenarios is the same, not necessarily the original ratio. Therefore, my original solution of x = 5 is correct.

Another check: Let me think if there's any constraints on x. For example, in the second scenario, Nate has 12 - x dogs. Since you can't have negative dogs, 12 - x must be greater than 0. Therefore, x < 12. Since x = 5, that's okay. Also, in the first scenario, Denali has 16 + 4x dogs. Since x is positive, that's fine. So, x = 5 is valid.

Therefore, I think the answer is 5.

But let me just go through the algebra again to make sure I didn't make any mistakes.

Original equation:

(16 + 4x)/12 = (16 + x)/(12 - x)

Cross-multiplying:

(16 + 4x)(12 - x) = 12(16 + x)

Expanding left side:

16*12 - 16x + 4x*12 - 4x^2 = 192 -16x + 48x -4x² = 192 + 32x -4x²

Right side: 12*16 + 12x = 192 + 12x

Set equal:

192 + 32x -4x² = 192 + 12x

Subtract 192:

32x -4x² = 12x

Subtract 12x:

20x -4x² = 0

Factor:

-4x² +20x = 0 => 4x² -20x = 0 => 4x(x -5) = 0 => x=0 or x=5. Correct.

Yes, that's correct. So, x=5 is the answer.

Therefore, I think the answer is 5.

**Final Answer**
The value of $ x $ is \boxed{5}.

To solve the problem, we are given that Denali and Nate are responsible for walking a certain number of dogs, and under a new policy, they can be assigned or reassigned dogs in groups of $ x $ dogs. The key condition is that the **ratio of Denali's pay to Nate's pay** remains the same in two different scenarios:

1. **Scenario 1**: Denali walks $ 4x $ more dogs, while Nate remains at 12 dogs.
2. **Scenario 2**: $ x $ of Nate's dogs are reassigned to Denali.

We are to find the value of $ x $, given that $ x \neq 0 $.

---

### Step 1: Express the Pay Ratios

Assuming pay is directly proportional to the number of dogs walked, the ratio of their pay is the same as the ratio of the number of dogs they walk.

- **Original scenario**: Denali walks 16 dogs, Nate walks 12 dogs → ratio = $ \frac{16}{12} = \frac{4}{3} $
- **Scenario 1**: Denali walks $ 16 + 4x $, Nate walks 12 → ratio = $ \frac{16 + 4x}{12} $
- **Scenario 2**: Denali walks $ 16 + x $, Nate walks $ 12 - x $ → ratio = $ \frac{16 + x}{12 - x} $

We are told that the ratio is the same in both scenarios, so we set the two expressions equal:

$$
\frac{16 + 4x}{12} = \frac{16 + x}{12 - x}
$$

---

### Step 2: Solve the Equation

Cross-multiply to eliminate the denominators:

$$
(16 + 4x)(12 - x) = 12(16 + x)
$$

Expand both sides:

**Left side:**
$$
(16 + 4x)(12 - x) = 16 \cdot 12 - 16x + 4x \cdot 12 - 4x^2 = 192 - 16x + 48x - 4x^2 = 192 + 32x - 4x^2
$$

**Right side:**
$$
12(16 + x) = 192 + 12x
$$

Now equate both sides:

$$
192 + 32x - 4x^2 = 192 + 12x
$$

Subtract 192 from both sides:

$$
32x - 4x^2 = 12x
$$

Subtract $ 12x $ from both sides:

$$
20x - 4x^2 = 0
$$

Factor:

$$
4x(5 - x) = 0
$$

So, the solutions are:

$$
x = 0 \quad \text{or} \quad x = 5
$$

Since $ x \neq 0 $, the only valid solution is:

$$
x = 5
$$

---

### Step 3: Verify the Solution

Let’s verify that the ratio is the same in both scenarios when $ x = 5 $:

- **Scenario 1**: Denali walks $ 16 + 4(5) = 36 $, Nate walks 12 → ratio = $ \frac{36}{12} = 3 $
- **Scenario 2**: Denali walks $ 16 + 5 = 21 $, Nate walks $ 12 - 5 = 7 $ → ratio = $ \frac{21}{7} = 3 $

Both ratios are equal, confirming that the solution is correct.

---

### Final Answer

$$
\boxed{5}
$$

Solution:
Rewriting the sentence "the ratio of Denali's pay to Nate's pay would be the same if Denali started walking $4x$ more dogs and Nate stayed at $12$ dogs or if $x$ of Nate's dogs were reassigned to Denali" as an equation, we have \[\frac{16+4x}{12}=\frac{16+x}{12-x}.\]Clearing denominators, \begin{align*}
(16+4x)(12-x)&=(16+x)(12)\quad \Rightarrow\\
192-16x+48x-4x^2&=192+12x\quad \Rightarrow\\
32x-4x^2&=12x\quad \Rightarrow\\
0&=4x^2-20x\quad \Rightarrow\\
0&=4x(x-5).
\end{align*}Because $x$ cannot be $0$, $x=\boxed{5}$.