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

Therefore, p(x) = [q(x) + x]/(x² - 1)

But q(x) is equal to the product (x - 2)...(x - 7)(x - r). Therefore, when we plug in x = 8, we have:

Numerator is q(8) + 8 = [product terms] + 8. Wait, but earlier I said that q(8) = product terms, which is (8 - 2)...(8 - 7)(8 - r). Then adding 8? Wait, no. Wait, p(x) = [q(x) + x]/(x² - 1). Therefore, when x = 8, p(8) = [q(8) + 8]/(8² - 1). But q(8) is equal to (8 - 2)...(8 - 7)(8 - r). Therefore, yes, the numerator is q(8) + 8. Wait, but earlier I thought that q(8) is equal to product*(8 - r). Wait, but earlier when I calculated q(1), I had q(1) = product*(1 - r). So yes, in general, q(x) = product*(x - r). Therefore, q(8) = product*(8 - r). Therefore, the numerator is product*(8 - r) + 8. Wait, but earlier I thought that product*(8 - r) was 5039? Wait, let me check again.

Wait, product of (x - 2)...(x - 7) evaluated at x = 8 is 6*5*4*3*2*1 = 720. Then multiplied by (8 - r) which is 8 - 721/720 = 5039/720. Therefore, q(8) = 720*(5039/720) = 5039. Then adding 8 gives 5039 + 8 = 5047. Then divide by 63. So p(8) = 5047 / 63. Let me check if this reduces.

Let me compute 5047 divided by 63. Let me see:

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. Let me check that.

Yes, because 63 is 7*9, and 5047 is 7*721. Therefore, cancelling 7, we get 721/9. So p(8) = 721/9. Let me check if that's correct.

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

First, we defined q(x) = (x² - 1)p(x) - x. Then, since p(n) = n/(n² - 1) for n = 2,...,7, then q(n) = 0 for those n. Therefore, q(x) has roots at 2,...,7, so q(x) = (x - 2)...(x - 7)(x - r). Then, we found r by evaluating q(1) = -1, which led us to r = 721/720. Then, to compute p(8), we compute [q(8) + 8]/(8² - 1). q(8) is (8 - 2)...(8 - 7)(8 - r) = 6*5*4*3*2*1*(8 - r) = 720*(8 - r). Then 8 - r = 8 - 721/720 = (5760 - 721)/720 = 5039/720. Therefore, q(8) = 720*(5039/720) = 5039. Then adding 8 gives 5039 + 8 = 5047. Divided by 63, which is 8² - 1. Then 5047 / 63 simplifies to 721/9. Let me check 721 divided by 9: 9*80 = 720, so 721/9 = 80 + 1/9. So that's 80.111..., which seems okay.

But let me check if there's another way to verify this answer. Maybe by considering another value?

Alternatively, maybe I can check if the polynomial p(x) constructed this way actually satisfies the original conditions? But that might be complicated. Alternatively, maybe there's a different approach.

Alternatively, think about the problem in terms of finite differences or something else? But since it's a degree 5 polynomial, maybe not. Alternatively, consider that the function f(x) = x/(x² - 1) can be written as [1/(x - 1)] - [1/(x + 1)]? Let me check:

Let me compute 1/(x - 1) - 1/(x + 1) = [ (x + 1) - (x - 1) ] / (x² - 1) ) = [2]/(x² - 1). Therefore, f(x) = x/(x² - 1) = [1/(x - 1) - 1/(x + 1)] / 2. Wait, so f(x) = [1/(x - 1) - 1/(x + 1)] / 2. Interesting. So maybe that helps?

But how? Maybe if I consider that p(x) agrees with f(x) at x = 2,...,7. So maybe we can think of p(x) as some combination of 1/(x - 1) and 1/(x + 1), but since p(x) is a polynomial, and 1/(x - 1) and 1/(x + 1) are not polynomials, but maybe we can use some interpolation?

Alternatively, since we have q(x) = (x² - 1)p(x) - x = (x - 2)...(x - 7)(x - r), and we found r = 721/720, then maybe we can use that to find p(8). Which is what I did earlier. So maybe that's correct.

Alternatively, maybe there's a mistake in the calculation of r? Let me check again.

Earlier, I calculated q(1) = -1. Then, q(1) = product*(1 - r) = 720*(1 - r) = -1. Therefore, 1 - r = -1/720 => r = 1 + 1/720 = 721/720. That seems correct.

Alternatively, maybe I can check with another value. For example, let me check x = 0. Wait, but we don't know p(0). However, maybe we can compute q(0) in two ways?

q(0) = (0² - 1)p(0) - 0 = -p(0). On the other hand, q(0) = (0 - 2)(0 - 3)...(0 - 7)(0 - r) = (-2)(-3)(-4)(-5)(-6)(-7)(-r). Let me compute that:

First, the product of (-2)(-3)(-4)(-5)(-6)(-7). Let me count the number of negative signs: there are 6 terms, each negative. So (-1)^6 * (2*3*4*5*6*7) = 1 * 5040. Then multiplied by (-r): 5040*(-r). Therefore, q(0) = -5040*r. Therefore, -p(0) = -5040*r => p(0) = 5040*r. Since r = 721/720, then p(0) = 5040*(721/720) = 7*721 = 5047. Wait, interesting. So p(0) = 5047. But how does that help?

Alternatively, maybe not helpful. But maybe if I check with another value? For example, x = 1. Wait, we already used x = 1 to find r. So maybe that's okay.

Alternatively, maybe I can check if p(8) = 721/9 is correct by considering the original equation. Let me compute 721 divided by 9. 9*80 = 720, so 721/9 = 80 + 1/9 ≈ 80.111...

Alternatively, let me think about the original function f(x) = x/(x² - 1). At x = 8, f(8) = 8/(64 - 1) = 8/63 ≈ 0.12698... But p(8) is 721/9 ≈ 80.111... which is way larger. But since p(x) is a polynomial that agrees with f(x) at points 2 through 7, but differs elsewhere. So it's possible for p(8) to be different. But maybe I can check with another approach?

Alternatively, think about constructing the polynomial p(x). Since p(x) interpolates f(x) at 6 points, and we have a degree 5 polynomial. So maybe using the method of finite differences? But that might be tedious.

Alternatively, think about the fact that p(x) - f(x) has roots at 2,...,7, but since f(x) isn't a polynomial, that might not help. Wait, but earlier we constructed q(x) which is a polynomial, so that seems better.

Alternatively, maybe there's a mistake in assuming that q(x) is equal to that product? Let me check again.

We have q(x) = (x² - 1)p(x) - x. Since p(x) is degree 5, then (x² - 1)p(x) is degree 7, and subtracting x doesn't change the degree, so q(x) is degree 7. Then, since q(x) has roots at 2,...,7, which are 6 roots, so we can write q(x) = (x - 2)...(x - 7)(ax + b). Wait, but earlier I thought it was (x - 2)...(x - 7)(x - r). But if it's degree 7, then it's a degree 7 polynomial, so it can be written as (x - 2)...(x - 7)(linear term). So maybe I should have written it as (x - 2)...(x - 7)(x - r), which is the same as before. So that's okay.

But then, when I evaluated q(1), I found r. So that seems okay.

Alternatively, maybe I can think of q(x) as (x - 2)...(x - 7)(x - r). Then, since q(x) = (x² - 1)p(x) - x, then if I can find another equation to solve for r? But we only have one equation from q(1) = -1. But since we have a degree 7 polynomial, and we have 6 roots, so only one unknown, which is r. Therefore, that's sufficient. Therefore, the calculation seems okay.

Therefore, I think my answer of 721/9 is correct. Let me check if 721/9 reduces further. 721 divided by... Let me check if 721 is divisible by 3: 7 + 2 + 1 = 10, which is not divisible by 3. Divided by 7? 7*100 = 700, 721 - 700 = 21, which is 7*3. Therefore, 721 = 7*103. Wait, 7*100 = 700, 7*3 = 21, so 7*103 = 721. Therefore, 721/9 = (7*103)/9. Since 103 is a prime number? Let me check. 103 divided by 2, no. 3? 1+0+3=4, not divisible by 3. 5? No. 7? 7*14=98, 103-98=5, not divisible by 7. 11? 11*9=99, 103-99=4. So yes, 103 is prime. Therefore, 721/9 is reduced.

But let me check if 721/9 is the correct answer. Alternatively, maybe I made a mistake in the calculation of q(8) + 8?

Wait, let me recompute:

q(8) = (8 - 2)(8 - 3)(8 - 4)(8 - 5)(8 - 6)(8 - 7)(8 - r) = 6*5*4*3*2*1*(8 - r) = 720*(8 - r). Then 8 - r = 8 - 721/720 = (8*720 - 721)/720 = (5760 - 721)/720 = 5039/720. Therefore, q(8) = 720*(5039/720) = 5039. Then adding 8 gives 5039 + 8 = 5047. Then divide by 63. 5047 / 63. As before, 63*80 = 5040, so 5047 = 63*80 + 7, so 5047/63 = 80 + 7/63 = 80 + 1/9 = 721/9. Yes, because 80*9 = 720, so 720 + 1 = 721. Therefore, 721/9 is correct.

Therefore, I think the answer is 721/9. Let me check if that's the simplest form. Yes, as discussed earlier. Therefore, I think that is the correct answer.

But just to be thorough, let me check with another approach. Let me consider that p(x) is a degree 5 polynomial. Let me consider the general form of p(x). But that might be too tedious. Alternatively, think about the difference between p(x) and f(x). Wait, but f(x) is not a polynomial. However, since p(x) agrees with f(x) at 6 points, maybe we can use some kind of interpolation formula?

Alternatively, think about the fact that p(x) - f(x) has zeros at x = 2,...,7, but since f(x) is not a polynomial, this isn't directly helpful. However, we already used that idea to construct q(x). So I think that approach was correct.

Alternatively, think about the problem in terms of constructing a polynomial that interpolates those points. But with Lagrange interpolation, the formula would be complicated, but maybe we can use the concept of finite differences or something else?

Alternatively, think about the fact that p(x) is a degree 5 polynomial, so if we can write p(x) as a combination of basis polynomials. But again, that might not be easier.

Alternatively, think about the fact that the problem is similar to the one where you have a polynomial passing through certain points and you need to find its value at another point, often using the concept of constructing a difference polynomial. Which is exactly what I did with q(x). Therefore, I think my approach is correct.

Therefore, I think the answer is 721/9. Therefore, I will go with that.

**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: Construct 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 $ n = 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 $ r $

We can find $ r $ by evaluating $ q(x) $ at a point not in the set $ \{2, 3, 4, 5, 6, 7\} $. A natural choice is $ x = 1 $, since $ x^2 - 1 = 0 $ at $ x = 1 $, and thus $ q(1) = -1 $, as shown below:

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

On the other hand, using the factored form:

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

Compute the product of the first six terms:

$$
(-1)(-2)(-3)(-4)(-5)(-6) = (-1)^6 \cdot 6! = 720
$$

So:

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

---

### Step 3: Use the Expression for $ p(x) $

Recall that:

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

Now, plug in $ x = 8 $:

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

Compute $ q(8) $:

$$
q(8) = (8 - 2)(8 - 3)(8 - 4)(8 - 5)(8 - 6)(8 - 7)(8 - r)
$$

$$
= 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 \cdot \left(8 - \frac{721}{720}\right)
= 720 \cdot \left(\frac{5760 - 721}{720}\right)
= 720 \cdot \frac{5039}{720} = 5039
$$

So:

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

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

Therefore, p(x) = [q(x) + x]/(x² - 1)

But q(x) is equal to the product (x - 2)...(x - 7)(x - r). Therefore, when we plug in x = 8, we have:

Numerator is q(8) + 8 = [product terms] + 8. Wait, but earlier I said that q(8) = product terms, which is (8 - 2)...(8 - 7)(8 - r). Then adding 8? Wait, no. Wait, p(x) = [q(x) + x]/(x² - 1). Therefore, when x = 8, p(8) = [q(8) + 8]/(8² - 1). But q(8) is equal to (8 - 2)...(8 - 7)(8 - r). Therefore, yes, the numerator is q(8) + 8. Wait, but earlier I thought that q(8) is equal to product*(8 - r). Wait, but earlier when I calculated q(1), I had q(1) = product*(1 - r). So yes, in general, q(x) = product*(x - r). Therefore, q(8) = product*(8 - r). Therefore, the numerator is product*(8 - r) + 8. Wait, but earlier I thought that product*(8 - r) was 5039? Wait, let me check again.

Wait, product of (x - 2)...(x - 7) evaluated at x = 8 is 6*5*4*3*2*1 = 720. Then multiplied by (8 - r) which is 8 - 721/720 = 5039/720. Therefore, q(8) = 720*(5039/720) = 5039. Then adding 8 gives 5039 + 8 = 5047. Then divide by 63. So p(8) = 5047 / 63. Let me check if this reduces.

Let me compute 5047 divided by 63. Let me see:

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. Let me check that.

Yes, because 63 is 7*9, and 5047 is 7*721. Therefore, cancelling 7, we get 721/9. So p(8) = 721/9. Let me check if that's correct.

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

First, we defined q(x) = (x² - 1)p(x) - x. Then, since p(n) = n/(n² - 1) for n = 2,...,7, then q(n) = 0 for those n. Therefore, q(x) has roots at 2,...,7, so q(x) = (x - 2)...(x - 7)(x - r). Then, we found r by evaluating q(1) = -1, which led us to r = 721/720. Then, to compute p(8), we compute [q(8) + 8]/(8² - 1). q(8) is (8 - 2)...(8 - 7)(8 - r) = 6*5*4*3*2*1*(8 - r) = 720*(8 - r). Then 8 - r = 8 - 721/720 = (5760 - 721)/720 = 5039/720. Therefore, q(8) = 720*(5039/720) = 5039. Then adding 8 gives 5039 + 8 = 5047. Divided by 63, which is 8² - 1. Then 5047 / 63 simplifies to 721/9. Let me check 721 divided by 9: 9*80 = 720, so 721/9 = 80 + 1/9. So that's 80.111..., which seems okay.

But let me check if there's another way to verify this answer. Maybe by considering another value?

Alternatively, maybe I can check if the polynomial p(x) constructed this way actually satisfies the original conditions? But that might be complicated. Alternatively, maybe there's a different approach.

Alternatively, think about the problem in terms of finite differences or something else? But since it's a degree 5 polynomial, maybe not. Alternatively, consider that the function f(x) = x/(x² - 1) can be written as [1/(x - 1)] - [1/(x + 1)]? Let me check:

Let me compute 1/(x - 1) - 1/(x + 1) = [ (x + 1) - (x - 1) ] / (x² - 1) ) = [2]/(x² - 1). Therefore, f(x) = x/(x² - 1) = [1/(x - 1) - 1/(x + 1)] / 2. Wait, so f(x) = [1/(x - 1) - 1/(x + 1)] / 2. Interesting. So maybe that helps?

But how? Maybe if I consider that p(x) agrees with f(x) at x = 2,...,7. So maybe we can think of p(x) as some combination of 1/(x - 1) and 1/(x + 1), but since p(x) is a polynomial, and 1/(x - 1) and 1/(x + 1) are not polynomials, but maybe we can use some interpolation?

Alternatively, since we have q(x) = (x² - 1)p(x) - x = (x - 2)...(x - 7)(x - r), and we found r = 721/720, then maybe we can use that to find p(8). Which is what I did earlier. So maybe that's correct.

Alternatively, maybe there's a mistake in the calculation of r? Let me check again.

Earlier, I calculated q(1) = -1. Then, q(1) = product*(1 - r) = 720*(1 - r) = -1. Therefore, 1 - r = -1/720 => r = 1 + 1/720 = 721/720. That seems correct.

Alternatively, maybe I can check with another value. For example, let me check x = 0. Wait, but we don't know p(0). However, maybe we can compute q(0) in two ways?

q(0) = (0² - 1)p(0) - 0 = -p(0). On the other hand, q(0) = (0 - 2)(0 - 3)...(0 - 7)(0 - r) = (-2)(-3)(-4)(-5)(-6)(-7)(-r). Let me compute that:

First, the product of (-2)(-3)(-4)(-5)(-6)(-7). Let me count the number of negative signs: there are 6 terms, each negative. So (-1)^6 * (2*3*4*5*6*7) = 1 * 5040. Then multiplied by (-r): 5040*(-r). Therefore, q(0) = -5040*r. Therefore, -p(0) = -5040*r => p(0) = 5040*r. Since r = 721/720, then p(0) = 5040*(721/720) = 7*721 = 5047. Wait, interesting. So p(0) = 5047. But how does that help?

Alternatively, maybe not helpful. But maybe if I check with another value? For example, x = 1. Wait, we already used x = 1 to find r. So maybe that's okay.

Alternatively, maybe I can check if p(8) = 721/9 is correct by considering the original equation. Let me compute 721 divided by 9. 9*80 = 720, so 721/9 = 80 + 1/9 ≈ 80.111...

Alternatively, let me think about the original function f(x) = x/(x² - 1). At x = 8, f(8) = 8/(64 - 1) = 8/63 ≈ 0.12698... But p(8) is 721/9 ≈ 80.111... which is way larger. But since p(x) is a polynomial that agrees with f(x) at points 2 through 7, but differs elsewhere. So it's possible for p(8) to be different. But maybe I can check with another approach?

Alternatively, think about constructing the polynomial p(x). Since p(x) interpolates f(x) at 6 points, and we have a degree 5 polynomial. So maybe using the method of finite differences? But that might be tedious.

Alternatively, think about the fact that p(x) - f(x) has roots at 2,...,7, but since f(x) isn't a polynomial, that might not help. Wait, but earlier we constructed q(x) which is a polynomial, so that seems better.

Alternatively, maybe there's a mistake in assuming that q(x) is equal to that product? Let me check again.

We have q(x) = (x² - 1)p(x) - x. Since p(x) is degree 5, then (x² - 1)p(x) is degree 7, and subtracting x doesn't change the degree, so q(x) is degree 7. Then, since q(x) has roots at 2,...,7, which are 6 roots, so we can write q(x) = (x - 2)...(x - 7)(ax + b). Wait, but earlier I thought it was (x - 2)...(x - 7)(x - r). But if it's degree 7, then it's a degree 7 polynomial, so it can be written as (x - 2)...(x - 7)(linear term). So maybe I should have written it as (x - 2)...(x - 7)(x - r), which is the same as before. So that's okay.

But then, when I evaluated q(1), I found r. So that seems okay.

Alternatively, maybe I can think of q(x) as (x - 2)...(x - 7)(x - r). Then, since q(x) = (x² - 1)p(x) - x, then if I can find another equation to solve for r? But we only have one equation from q(1) = -1. But since we have a degree 7 polynomial, and we have 6 roots, so only one unknown, which is r. Therefore, that's sufficient. Therefore, the calculation seems okay.

Therefore, I think my answer of 721/9 is correct. Let me check if 721/9 reduces further. 721 divided by... Let me check if 721 is divisible by 3: 7 + 2 + 1 = 10, which is not divisible by 3. Divided by 7? 7*100 = 700, 721 - 700 = 21, which is 7*3. Therefore, 721 = 7*103. Wait, 7*100 = 700, 7*3 = 21, so 7*103 = 721. Therefore, 721/9 = (7*103)/9. Since 103 is a prime number? Let me check. 103 divided by 2, no. 3? 1+0+3=4, not divisible by 3. 5? No. 7? 7*14=98, 103-98=5, not divisible by 7. 11? 11*9=99, 103-99=4. So yes, 103 is prime. Therefore, 721/9 is reduced.

But let me check if 721/9 is the correct answer. Alternatively, maybe I made a mistake in the calculation of q(8) + 8?

Wait, let me recompute:

q(8) = (8 - 2)(8 - 3)(8 - 4)(8 - 5)(8 - 6)(8 - 7)(8 - r) = 6*5*4*3*2*1*(8 - r) = 720*(8 - r). Then 8 - r = 8 - 721/720 = (8*720 - 721)/720 = (5760 - 721)/720 = 5039/720. Therefore, q(8) = 720*(5039/720) = 5039. Then adding 8 gives 5039 + 8 = 5047. Then divide by 63. 5047 / 63. As before, 63*80 = 5040, so 5047 = 63*80 + 7, so 5047/63 = 80 + 7/63 = 80 + 1/9 = 721/9. Yes, because 80*9 = 720, so 720 + 1 = 721. Therefore, 721/9 is correct.

Therefore, I think the answer is 721/9. Let me check if that's the simplest form. Yes, as discussed earlier. Therefore, I think that is the correct answer.

But just to be thorough, let me check with another approach. Let me consider that p(x) is a degree 5 polynomial. Let me consider the general form of p(x). But that might be too tedious. Alternatively, think about the difference between p(x) and f(x). Wait, but f(x) is not a polynomial. However, since p(x) agrees with f(x) at 6 points, maybe we can use some kind of interpolation formula?

Alternatively, think about the fact that p(x) - f(x) has zeros at x = 2,...,7, but since f(x) is not a polynomial, this isn't directly helpful. However, we already used that idea to construct q(x). So I think that approach was correct.

Alternatively, think about the problem in terms of constructing a polynomial that interpolates those points. But with Lagrange interpolation, the formula would be complicated, but maybe we can use the concept of finite differences or something else?

Alternatively, think about the fact that p(x) is a degree 5 polynomial, so if we can write p(x) as a combination of basis polynomials. But again, that might not be easier.

Alternatively, think about the fact that the problem is similar to the one where you have a polynomial passing through certain points and you need to find its value at another point, often using the concept of constructing a difference polynomial. Which is exactly what I did with q(x). Therefore, I think my approach is correct.

Therefore, I think the answer is 721/9. Therefore, I will go with that.

**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: Construct 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 $ n = 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 $ r $

We can find $ r $ by evaluating $ q(x) $ at a point not in the set $ \{2, 3, 4, 5, 6, 7\} $. A natural choice is $ x = 1 $, since $ x^2 - 1 = 0 $ at $ x = 1 $, and thus $ q(1) = -1 $, as shown below:

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

On the other hand, using the factored form:

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

Compute the product of the first six terms:

$$
(-1)(-2)(-3)(-4)(-5)(-6) = (-1)^6 \cdot 6! = 720
$$

So:

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

---

### Step 3: Use the Expression for $ p(x) $

Recall that:

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

Now, plug in $ x = 8 $:

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

Compute $ q(8) $:

$$
q(8) = (8 - 2)(8 - 3)(8 - 4)(8 - 5)(8 - 6)(8 - 7)(8 - r)
$$

$$
= 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 \cdot \left(8 - \frac{721}{720}\right)
= 720 \cdot \left(\frac{5760 - 721}{720}\right)
= 720 \cdot \frac{5039}{720} = 5039
$$

So:

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

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, to find all divisors, we take exponents of primes from 0 up to their maximum in the factorization.

For 2, exponents can be 0, 1, or 2. For 71, exponents can be 0 or 1.

So the divisors are:

2⁰ × 71⁰ = 1

2¹ × 71⁰ = 2

2² × 71⁰ = 4

2⁰ × 71¹ = 71

2¹ × 71¹ = 142

2² × 71¹ = 284

So the divisors of 284 are 1, 2, 4, 71, 142, 284.

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

Now, I need to sum these up. Let me add them step by step.

Starting with 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 next part of the problem is to find the sum of the proper divisors of 220.

So now I need to find the proper divisors of 220 and add them up.

Again, let me factorize 220 to find all its divisors.

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

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

To find all divisors, we take exponents for each prime:

For 2: 0, 1, 2

For 5: 0, 1

For 11: 0, 1

So the number of divisors is (2+1)(1+1)(1+1) = 3×2×2=12 divisors. Let me list them all:

Starting with 2^0:

- 5^0 × 11^0 = 1

- 5^0 × 11^1 = 11

- 5^1 × 11^0 = 5

- 5^1 × 11^1 = 55

Then 2^1:

- 2^1 × 5^0 × 11^0 = 2

- 2^1 × 5^0 × 11^1 = 22

- 2^1 × 5^1 × 11^0 = 10

- 2^1 × 5^1 × 11^1 = 110

Then 2^2:

- 2^2 × 5^0 × 11^0 = 4

- 2^2 × 5^0 × 11^1 = 44

- 2^2 × 5^1 × 11^0 = 20

- 2^2 × 5^1 × 11^1 = 220

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

Now, proper divisors exclude 220, so we remove that. Therefore, the proper divisors are: 1, 11, 5, 55, 2, 22, 10, 110, 4, 44, 20.

Now, let me sum these up. Let me group them to make it easier.

First, let me list them again:

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

Wait, let me check if I have all of them. Original list was:

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

Yes, that's 11 numbers. Let me count again:

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

Now, adding them up step by step.

Start with 1 + 2 = 3

3 + 4 = 7

7 + 5 = 12

12 + 10 = 22

22 + 11 = 33

33 + 20 = 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, 11, 20, 22, 44, 55, 110.

Let me add them in pairs or groups:

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

Alternatively, maybe group them:

Group 1: 1 + 2 + 4 = 7

Group 2: 5 + 10 = 15

Group 3: 11 + 20 = 31

Group 4: 22 + 44 = 66

Group 5: 55 + 110 = 165

Now add these groups: 7 + 15 = 22; 22 + 31 = 53; 53 + 66 = 119; 119 + 165 = 284.

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

Therefore, the problem asks for the sum of the proper divisors of the sum of the proper divisors of 284.

We found that the sum of the proper divisors of 284 is 220, and then the sum of the proper divisors of 220 is 284.

Therefore, the answer is 284.

But let me just cross-verify this because 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 makes sense. So the sum of proper divisors of 284 is 220, and the sum of proper divisors of 220 is 284.

Therefore, the answer is 284.

But let me just check once again my calculation for the sum of proper divisors of 220. Maybe I made a mistake in listing the divisors? Let me check again.

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

Yes, that's correct. So removing 220, the proper divisors are the first 11 numbers.

Adding them up: 1+2+4+5+10+11+20+22+44+55+110.

Alternatively, maybe I can use another method. For example, using the formula for sum of divisors.

The sum of all divisors of a number is given by the product of (p^(k+1) - 1)/(p - 1) for each prime factor p with exponent k.

So for 220, which is 2² × 5¹ × 11¹, the sum of all divisors is:

(2³ - 1)/(2 - 1) × (5² - 1)/(5 - 1) × (11² - 1)/(11 - 1)

Calculating each part:

For 2²: (2³ - 1)/1 = 8 - 1 = 7

For 5¹: (5² - 1)/4 = (25 - 1)/4 = 24/4 = 6

For 11¹: (11² - 1)/10 = (121 - 1)/10 = 120/10 = 12

Multiply them all together: 7 × 6 × 12.

First, 7 × 6 = 42. Then 42 × 12. Let me compute that: 42 × 10 = 420, 42 × 2 = 84, so total 420 + 84 = 504.

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

Yes! That confirms it. So the sum of proper divisors of 220 is indeed 284.

Therefore, the answer is 284.

So, to recap:

Sum of proper divisors of 284 is 220.

Sum of proper divisors of 220 is 284.

Therefore, the answer is 284.

I think that's solid. I checked it two different ways: by listing the divisors and adding them up, and by using the formula for the sum of divisors. Both methods gave me the same result, so I feel confident that 284 is the correct answer.

**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**:

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

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

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

Now, remove 284 (since it's the number itself), and we are left with the **proper divisors**:

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

Now, compute their **sum**:

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

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

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

Remove 220 (the number itself), and we are left with the **proper divisors**:

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

Now, compute their **sum**:

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

---

### Final Step: Conclusion

We were asked to find the **sum of the proper divisors of the sum of the proper divisors of 284**.

- The **sum of the proper divisors of 284** is **220**.
- The **sum of the proper divisors of 220** is **284**.

Thus, the final answer 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, to find all divisors, we take exponents of primes from 0 up to their maximum in the factorization.

For 2, exponents can be 0, 1, or 2. For 71, exponents can be 0 or 1.

So the divisors are:

2⁰ × 71⁰ = 1

2¹ × 71⁰ = 2

2² × 71⁰ = 4

2⁰ × 71¹ = 71

2¹ × 71¹ = 142

2² × 71¹ = 284

So the divisors of 284 are 1, 2, 4, 71, 142, 284.

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

Now, I need to sum these up. Let me add them step by step.

Starting with 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 next part of the problem is to find the sum of the proper divisors of 220.

So now I need to find the proper divisors of 220 and add them up.

Again, let me factorize 220 to find all its divisors.

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

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

To find all divisors, we take exponents for each prime:

For 2: 0, 1, 2

For 5: 0, 1

For 11: 0, 1

So the number of divisors is (2+1)(1+1)(1+1) = 3×2×2=12 divisors. Let me list them all:

Starting with 2^0:

- 5^0 × 11^0 = 1

- 5^0 × 11^1 = 11

- 5^1 × 11^0 = 5

- 5^1 × 11^1 = 55

Then 2^1:

- 2^1 × 5^0 × 11^0 = 2

- 2^1 × 5^0 × 11^1 = 22

- 2^1 × 5^1 × 11^0 = 10

- 2^1 × 5^1 × 11^1 = 110

Then 2^2:

- 2^2 × 5^0 × 11^0 = 4

- 2^2 × 5^0 × 11^1 = 44

- 2^2 × 5^1 × 11^0 = 20

- 2^2 × 5^1 × 11^1 = 220

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

Now, proper divisors exclude 220, so we remove that. Therefore, the proper divisors are: 1, 11, 5, 55, 2, 22, 10, 110, 4, 44, 20.

Now, let me sum these up. Let me group them to make it easier.

First, let me list them again:

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

Wait, let me check if I have all of them. Original list was:

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

Yes, that's 11 numbers. Let me count again:

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

Now, adding them up step by step.

Start with 1 + 2 = 3

3 + 4 = 7

7 + 5 = 12

12 + 10 = 22

22 + 11 = 33

33 + 20 = 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, 11, 20, 22, 44, 55, 110.

Let me add them in pairs or groups:

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

Alternatively, maybe group them:

Group 1: 1 + 2 + 4 = 7

Group 2: 5 + 10 = 15

Group 3: 11 + 20 = 31

Group 4: 22 + 44 = 66

Group 5: 55 + 110 = 165

Now add these groups: 7 + 15 = 22; 22 + 31 = 53; 53 + 66 = 119; 119 + 165 = 284.

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

Therefore, the problem asks for the sum of the proper divisors of the sum of the proper divisors of 284.

We found that the sum of the proper divisors of 284 is 220, and then the sum of the proper divisors of 220 is 284.

Therefore, the answer is 284.

But let me just cross-verify this because 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 makes sense. So the sum of proper divisors of 284 is 220, and the sum of proper divisors of 220 is 284.

Therefore, the answer is 284.

But let me just check once again my calculation for the sum of proper divisors of 220. Maybe I made a mistake in listing the divisors? Let me check again.

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

Yes, that's correct. So removing 220, the proper divisors are the first 11 numbers.

Adding them up: 1+2+4+5+10+11+20+22+44+55+110.

Alternatively, maybe I can use another method. For example, using the formula for sum of divisors.

The sum of all divisors of a number is given by the product of (p^(k+1) - 1)/(p - 1) for each prime factor p with exponent k.

So for 220, which is 2² × 5¹ × 11¹, the sum of all divisors is:

(2³ - 1)/(2 - 1) × (5² - 1)/(5 - 1) × (11² - 1)/(11 - 1)

Calculating each part:

For 2²: (2³ - 1)/1 = 8 - 1 = 7

For 5¹: (5² - 1)/4 = (25 - 1)/4 = 24/4 = 6

For 11¹: (11² - 1)/10 = (121 - 1)/10 = 120/10 = 12

Multiply them all together: 7 × 6 × 12.

First, 7 × 6 = 42. Then 42 × 12. Let me compute that: 42 × 10 = 420, 42 × 2 = 84, so total 420 + 84 = 504.

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

Yes! That confirms it. So the sum of proper divisors of 220 is indeed 284.

Therefore, the answer is 284.

So, to recap:

Sum of proper divisors of 284 is 220.

Sum of proper divisors of 220 is 284.

Therefore, the answer is 284.

I think that's solid. I checked it two different ways: by listing the divisors and adding them up, and by using the formula for the sum of divisors. Both methods gave me the same result, so I feel confident that 284 is the correct answer.

**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**:

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

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

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

Now, remove 284 (since it's the number itself), and we are left with the **proper divisors**:

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

Now, compute their **sum**:

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

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

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

Remove 220 (the number itself), and we are left with the **proper divisors**:

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

Now, compute their **sum**:

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

---

### Final Step: Conclusion

We were asked to find the **sum of the proper divisors of the sum of the proper divisors of 284**.

- The **sum of the proper divisors of 284** is **220**.
- The **sum of the proper divisors of 220** is **284**.

Thus, the final answer 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 = z - (c_real + c_imag). So:

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

So real part: 2 + √2 - 2 = √2

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

So z - c = √2 - 3√2 i. Which can be written as √2(1 - 3i).

Okay, so that simplifies things. So z - c is √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, since multiplying by e^{iθ} can be represented as multiplying by a complex number with magnitude 1 and angle θ. So, if I have a complex number in the form a + bi, multiplying by e^{iπ/4} is equivalent to rotating it by π/4.

Alternatively, maybe I can write z - c in polar form and then multiply by e^{iπ/4}?

But maybe it's easier to just do the multiplication directly. Let me try that.

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

Multiplying by e^{iπ/4} = (√2/2 + i√2/2).

So, the product is √2*(1 - 3i)*(√2/2 + i√2/2).

Let me compute that step by step.

First, let me compute (1 - 3i)*(√2/2 + i√2/2). Let me factor out √2/2 from both terms:

(1 - 3i) * √2/2 * (1 + i).

So, that's √2/2 * (1 - 3i)(1 + i).

Let me compute (1 - 3i)(1 + i):

Multiply using FOIL:

First: 1*1 = 1

Outer: 1*i = i

Inner: -3i*1 = -3i

Last: -3i*i = -3i² = -3(-1) = 3

So adding them up: 1 + i - 3i + 3 = (1 + 3) + (i - 3i) = 4 - 2i

Therefore, (1 - 3i)(1 + i) = 4 - 2i

Therefore, the entire product is √2/2 * (4 - 2i) = √2/2 * 2*(2 - i) = √2*(2 - i)

Wait, let me check that again:

√2/2 * (4 - 2i) = √2/2 * 2*(2 - i) = √2*(2 - i). Yes, because 4 - 2i is 2*(2 - i).

So, that simplifies to √2*(2 - i).

But remember, this was multiplied by the original √2 from z - c. Wait, no. Wait, earlier I had:

Original z - c is √2*(1 - 3i). Then multiplied by e^{iπ/4} which is √2/2*(1 + i). So, the entire product is √2*(1 - 3i) * √2/2*(1 + i).

Which is (√2 * √2 / 2) * (1 - 3i)(1 + i) = (2 / 2) * (4 - 2i) = 1*(4 - 2i) = 4 - 2i.

Wait, that seems different from my previous step. Wait, let me check again.

Wait, z - c is √2*(1 - 3i). Then multiplying by e^{iπ/4} which is (√2/2 + i√2/2). So, the entire product is √2*(1 - 3i) * (√2/2 + i√2/2).

Let me compute that:

First, multiply √2*(√2/2) = (2)/2 = 1.

Then, √2*(i√2/2) = i*(2)/2 = i.

Then, the other terms: (1 - 3i)*(√2/2) and (1 - 3i)*(i√2/2). Wait, maybe I need to do it more carefully.

Alternatively, perhaps I should consider multiplying the two complex numbers:

Let me write z - c as √2*(1 - 3i) and e^{iπ/4} as (√2/2 + i√2/2).

So, multiplying them:

√2*(1 - 3i) * (√2/2 + i√2/2)

Let me compute the constants first: √2 * √2/2 = (2)/2 = 1.

Then, √2 * i√2/2 = i*(2)/2 = i.

But actually, this is not the correct way to multiply two complex numbers. Wait, no. Wait, the entire expression is:

Multiply √2*(1 - 3i) by (√2/2 + i√2/2).

So, distribute the multiplication:

√2*(1 - 3i)*(√2/2) + √2*(1 - 3i)*(i√2/2)

Compute each term:

First term: √2*(√2/2)*(1 - 3i) = (2/2)*(1 - 3i) = 1*(1 - 3i) = 1 - 3i

Second term: √2*(i√2/2)*(1 - 3i) = i*(2)/2*(1 - 3i) = i*(1 - 3i) = i - 3i² = i - 3(-1) = i + 3

Therefore, adding the two terms: (1 - 3i) + (i + 3) = (1 + 3) + (-3i + i) = 4 - 2i.

So, the product (z - c) * e^{iπ/4} is 4 - 2i.

Therefore, going back to the formula for w:

w = (z - c) * e^{iπ/4} + c = (4 - 2i) + c

But c is 2 - 3i. Therefore:

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

Wait, so is that the answer? Let me check again.

Wait, let me recap:

z - c = √2(1 - 3i)

Multiply by e^{iπ/4} gives 4 - 2i

Then add c = 2 - 3i:

4 - 2i + 2 - 3i = 6 - 5i.

So, w = 6 - 5i?

But let me check if that's correct. Maybe I made a mistake in the multiplication steps. Let me verify.

Alternatively, maybe I can approach this problem using vectors or coordinates. Let me try that method to cross-verify.

First, let me represent z and c as points in the complex plane.

z is (2 + √2, -3 - 3√2).

c is (2, -3).

So, the vector from c to z is z - c, which we already calculated as (√2, -3√2).

So, the vector from c to z is (√2, -3√2).

Now, 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, applying this to the vector (√2, -3√2):

New x-coordinate: (√2)(√2/2) + (-3√2)(-√2/2)

New y-coordinate: (√2)(√2/2) + (-3√2)(√2/2)

Let me compute the x-coordinate:

First term: √2 * √2 / 2 = 2 / 2 = 1

Second term: (-3√2) * (-√2)/2 = 3*(√2 * √2)/2 = 3*(2)/2 = 3

So total x-coordinate: 1 + 3 = 4

Now y-coordinate:

First term: √2 * √2 / 2 = 1

Second term: (-3√2) * √2 / 2 = -3*(2)/2 = -3

So total y-coordinate: 1 - 3 = -2

Therefore, the rotated vector is (4, -2).

Therefore, the new point w is c + rotated vector.

Since c is (2, -3), adding (4, -2) gives:

x: 2 + 4 = 6

y: -3 + (-2) = -5

Therefore, w is (6, -5), which corresponds to the complex number 6 - 5i.

So, same result as before. Therefore, seems like w = 6 - 5i.

But let me check once more with another approach. Maybe using complex numbers directly.

Alternatively, 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 + [ -3i - 3√2 i + 3i ] = √2 - 3√2 i.

Yes, same as before.

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

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

Let me compute this multiplication:

First, write √2 - 3√2 i as √2(1 - 3i).

Multiply by √2/2 + i√2/2:

√2(1 - 3i) * √2/2 (1 + i) = (√2 * √2 / 2) * (1 - 3i)(1 + i) = (2 / 2) * (1 - 3i)(1 + i) = 1 * (1 + i - 3i - 3i²) = 1*(1 - 2i + 3) = 1*(4 - 2i) = 4 - 2i.

Then add c = 2 - 3i: 4 - 2i + 2 - 3i = 6 - 5i.

Same result.

Therefore, seems like the answer is 6 - 5i.

But let me check with the Asymptote code given. The Asymptote code defines Z as (2 + sqrt(2), -3 - 3*sqrt(2)), which matches our z. Then C is (2, -3), which is c. Then W is rotate(45,C)*(Z).

So, according to the Asymptote code, rotating Z around C by 45 degrees.

If we compute coordinates, as we did, we get (6, -5). So, the answer should be 6 - 5i.

But let me check once more with another method.

Alternatively, using complex numbers, let me write z as a complex number:

z = (2 + √2) - (3 + 3√2)i

c = 2 - 3i

So, to rotate z around c by π/4, we can use the formula:

w = c + (z - c) * e^{iπ/4}

Which is exactly what we did.

Alternatively, maybe I can compute e^{iπ/4} as a complex number and multiply.

Let me write e^{iπ/4} = cos(π/4) + i sin(π/4) = √2/2 + i√2/2.

So, (z - c) is √2 - 3√2 i.

Multiply this by √2/2 + i√2/2:

Let me denote a = √2, b = -3√2, so the complex number is a + b i.

Multiply by √2/2 + i√2/2:

(a + b i)(√2/2 + i√2/2) = a*(√2/2) + a*(i√2/2) + b i*(√2/2) + b i*(i√2/2)

Compute each term:

First term: a*(√2/2) = √2*(√2)/2 = 2/2 = 1

Second term: a*(i√2/2) = √2*(i√2)/2 = i*(2)/2 = i

Third term: b i*(√2/2) = (-3√2)*i*(√2)/2 = (-3*2)/2 * i = -3i

Fourth term: b i*(i√2/2) = (-3√2)*i*(i√2)/2 = (-3√2 * √2)/2 * i² = (-3*2)/2 * (-1) = (-3)*(-1) = 3

So adding all terms:

First term: 1

Second term: +i

Third term: -3i

Fourth term: +3

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

Same result. Then adding c = 2 - 3i gives 4 - 2i + 2 - 3i = 6 - 5i.

Therefore, all methods lead to the same answer.

But just to be thorough, let me check if there's any chance I messed up the rotation direction. 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.

Alternatively, if it were clockwise, we would have used e^{-iπ/4}, but since it's counter-clockwise, we are correct.

Another check: Let me compute the magnitude of z - c and see if after rotation, the magnitude remains the same.

Original vector z - c is √2 - 3√2 i. Its magnitude is sqrt( (√2)^2 + (-3√2)^2 ) = sqrt(2 + 9*2) = sqrt(2 + 18) = sqrt(20) = 2*sqrt(5).

After rotation, the vector is 4 - 2i. Its magnitude is sqrt(4^2 + (-2)^2) = sqrt(16 + 4) = sqrt(20) = 2*sqrt(5). So same magnitude, which is correct.

Therefore, the rotation preserves the magnitude, which is consistent.

Therefore, I think the answer is correct.

So, the final answer is w = 6 - 5i.

**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

We are given:

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

---

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

Subtract $ c $ from $ z $:

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

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

---

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

We use the identity:

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

Now compute:

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

Factor out $ \sqrt{2} $ from both terms:

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

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

Now compute the product:

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

So:

$$
(z - c) \cdot e^{i\pi/4} = 4 - 2i
$$

---

### Step 4: Add back $ c $

Now compute:

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

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

---

### 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 = z - (c_real + c_imag). So:

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

So real part: 2 + √2 - 2 = √2

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

So z - c = √2 - 3√2 i. Which can be written as √2(1 - 3i).

Okay, so that simplifies things. So z - c is √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, since multiplying by e^{iθ} can be represented as multiplying by a complex number with magnitude 1 and angle θ. So, if I have a complex number in the form a + bi, multiplying by e^{iπ/4} is equivalent to rotating it by π/4.

Alternatively, maybe I can write z - c in polar form and then multiply by e^{iπ/4}?

But maybe it's easier to just do the multiplication directly. Let me try that.

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

Multiplying by e^{iπ/4} = (√2/2 + i√2/2).

So, the product is √2*(1 - 3i)*(√2/2 + i√2/2).

Let me compute that step by step.

First, let me compute (1 - 3i)*(√2/2 + i√2/2). Let me factor out √2/2 from both terms:

(1 - 3i) * √2/2 * (1 + i).

So, that's √2/2 * (1 - 3i)(1 + i).

Let me compute (1 - 3i)(1 + i):

Multiply using FOIL:

First: 1*1 = 1

Outer: 1*i = i

Inner: -3i*1 = -3i

Last: -3i*i = -3i² = -3(-1) = 3

So adding them up: 1 + i - 3i + 3 = (1 + 3) + (i - 3i) = 4 - 2i

Therefore, (1 - 3i)(1 + i) = 4 - 2i

Therefore, the entire product is √2/2 * (4 - 2i) = √2/2 * 2*(2 - i) = √2*(2 - i)

Wait, let me check that again:

√2/2 * (4 - 2i) = √2/2 * 2*(2 - i) = √2*(2 - i). Yes, because 4 - 2i is 2*(2 - i).

So, that simplifies to √2*(2 - i).

But remember, this was multiplied by the original √2 from z - c. Wait, no. Wait, earlier I had:

Original z - c is √2*(1 - 3i). Then multiplied by e^{iπ/4} which is √2/2*(1 + i). So, the entire product is √2*(1 - 3i) * √2/2*(1 + i).

Which is (√2 * √2 / 2) * (1 - 3i)(1 + i) = (2 / 2) * (4 - 2i) = 1*(4 - 2i) = 4 - 2i.

Wait, that seems different from my previous step. Wait, let me check again.

Wait, z - c is √2*(1 - 3i). Then multiplying by e^{iπ/4} which is (√2/2 + i√2/2). So, the entire product is √2*(1 - 3i) * (√2/2 + i√2/2).

Let me compute that:

First, multiply √2*(√2/2) = (2)/2 = 1.

Then, √2*(i√2/2) = i*(2)/2 = i.

Then, the other terms: (1 - 3i)*(√2/2) and (1 - 3i)*(i√2/2). Wait, maybe I need to do it more carefully.

Alternatively, perhaps I should consider multiplying the two complex numbers:

Let me write z - c as √2*(1 - 3i) and e^{iπ/4} as (√2/2 + i√2/2).

So, multiplying them:

√2*(1 - 3i) * (√2/2 + i√2/2)

Let me compute the constants first: √2 * √2/2 = (2)/2 = 1.

Then, √2 * i√2/2 = i*(2)/2 = i.

But actually, this is not the correct way to multiply two complex numbers. Wait, no. Wait, the entire expression is:

Multiply √2*(1 - 3i) by (√2/2 + i√2/2).

So, distribute the multiplication:

√2*(1 - 3i)*(√2/2) + √2*(1 - 3i)*(i√2/2)

Compute each term:

First term: √2*(√2/2)*(1 - 3i) = (2/2)*(1 - 3i) = 1*(1 - 3i) = 1 - 3i

Second term: √2*(i√2/2)*(1 - 3i) = i*(2)/2*(1 - 3i) = i*(1 - 3i) = i - 3i² = i - 3(-1) = i + 3

Therefore, adding the two terms: (1 - 3i) + (i + 3) = (1 + 3) + (-3i + i) = 4 - 2i.

So, the product (z - c) * e^{iπ/4} is 4 - 2i.

Therefore, going back to the formula for w:

w = (z - c) * e^{iπ/4} + c = (4 - 2i) + c

But c is 2 - 3i. Therefore:

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

Wait, so is that the answer? Let me check again.

Wait, let me recap:

z - c = √2(1 - 3i)

Multiply by e^{iπ/4} gives 4 - 2i

Then add c = 2 - 3i:

4 - 2i + 2 - 3i = 6 - 5i.

So, w = 6 - 5i?

But let me check if that's correct. Maybe I made a mistake in the multiplication steps. Let me verify.

Alternatively, maybe I can approach this problem using vectors or coordinates. Let me try that method to cross-verify.

First, let me represent z and c as points in the complex plane.

z is (2 + √2, -3 - 3√2).

c is (2, -3).

So, the vector from c to z is z - c, which we already calculated as (√2, -3√2).

So, the vector from c to z is (√2, -3√2).

Now, 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, applying this to the vector (√2, -3√2):

New x-coordinate: (√2)(√2/2) + (-3√2)(-√2/2)

New y-coordinate: (√2)(√2/2) + (-3√2)(√2/2)

Let me compute the x-coordinate:

First term: √2 * √2 / 2 = 2 / 2 = 1

Second term: (-3√2) * (-√2)/2 = 3*(√2 * √2)/2 = 3*(2)/2 = 3

So total x-coordinate: 1 + 3 = 4

Now y-coordinate:

First term: √2 * √2 / 2 = 1

Second term: (-3√2) * √2 / 2 = -3*(2)/2 = -3

So total y-coordinate: 1 - 3 = -2

Therefore, the rotated vector is (4, -2).

Therefore, the new point w is c + rotated vector.

Since c is (2, -3), adding (4, -2) gives:

x: 2 + 4 = 6

y: -3 + (-2) = -5

Therefore, w is (6, -5), which corresponds to the complex number 6 - 5i.

So, same result as before. Therefore, seems like w = 6 - 5i.

But let me check once more with another approach. Maybe using complex numbers directly.

Alternatively, 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 + [ -3i - 3√2 i + 3i ] = √2 - 3√2 i.

Yes, same as before.

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

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

Let me compute this multiplication:

First, write √2 - 3√2 i as √2(1 - 3i).

Multiply by √2/2 + i√2/2:

√2(1 - 3i) * √2/2 (1 + i) = (√2 * √2 / 2) * (1 - 3i)(1 + i) = (2 / 2) * (1 - 3i)(1 + i) = 1 * (1 + i - 3i - 3i²) = 1*(1 - 2i + 3) = 1*(4 - 2i) = 4 - 2i.

Then add c = 2 - 3i: 4 - 2i + 2 - 3i = 6 - 5i.

Same result.

Therefore, seems like the answer is 6 - 5i.

But let me check with the Asymptote code given. The Asymptote code defines Z as (2 + sqrt(2), -3 - 3*sqrt(2)), which matches our z. Then C is (2, -3), which is c. Then W is rotate(45,C)*(Z).

So, according to the Asymptote code, rotating Z around C by 45 degrees.

If we compute coordinates, as we did, we get (6, -5). So, the answer should be 6 - 5i.

But let me check once more with another method.

Alternatively, using complex numbers, let me write z as a complex number:

z = (2 + √2) - (3 + 3√2)i

c = 2 - 3i

So, to rotate z around c by π/4, we can use the formula:

w = c + (z - c) * e^{iπ/4}

Which is exactly what we did.

Alternatively, maybe I can compute e^{iπ/4} as a complex number and multiply.

Let me write e^{iπ/4} = cos(π/4) + i sin(π/4) = √2/2 + i√2/2.

So, (z - c) is √2 - 3√2 i.

Multiply this by √2/2 + i√2/2:

Let me denote a = √2, b = -3√2, so the complex number is a + b i.

Multiply by √2/2 + i√2/2:

(a + b i)(√2/2 + i√2/2) = a*(√2/2) + a*(i√2/2) + b i*(√2/2) + b i*(i√2/2)

Compute each term:

First term: a*(√2/2) = √2*(√2)/2 = 2/2 = 1

Second term: a*(i√2/2) = √2*(i√2)/2 = i*(2)/2 = i

Third term: b i*(√2/2) = (-3√2)*i*(√2)/2 = (-3*2)/2 * i = -3i

Fourth term: b i*(i√2/2) = (-3√2)*i*(i√2)/2 = (-3√2 * √2)/2 * i² = (-3*2)/2 * (-1) = (-3)*(-1) = 3

So adding all terms:

First term: 1

Second term: +i

Third term: -3i

Fourth term: +3

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

Same result. Then adding c = 2 - 3i gives 4 - 2i + 2 - 3i = 6 - 5i.

Therefore, all methods lead to the same answer.

But just to be thorough, let me check if there's any chance I messed up the rotation direction. 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.

Alternatively, if it were clockwise, we would have used e^{-iπ/4}, but since it's counter-clockwise, we are correct.

Another check: Let me compute the magnitude of z - c and see if after rotation, the magnitude remains the same.

Original vector z - c is √2 - 3√2 i. Its magnitude is sqrt( (√2)^2 + (-3√2)^2 ) = sqrt(2 + 9*2) = sqrt(2 + 18) = sqrt(20) = 2*sqrt(5).

After rotation, the vector is 4 - 2i. Its magnitude is sqrt(4^2 + (-2)^2) = sqrt(16 + 4) = sqrt(20) = 2*sqrt(5). So same magnitude, which is correct.

Therefore, the rotation preserves the magnitude, which is consistent.

Therefore, I think the answer is correct.

So, the final answer is w = 6 - 5i.

**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

We are given:

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

---

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

Subtract $ c $ from $ z $:

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

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

---

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

We use the identity:

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

Now compute:

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

Factor out $ \sqrt{2} $ from both terms:

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

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

Now compute the product:

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

So:

$$
(z - c) \cdot e^{i\pi/4} = 4 - 2i
$$

---

### Step 4: Add back $ c $

Now compute:

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

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

---

### 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 $0$, $x=\boxed{5}$.