To solve this question, we need to find the largest positive integer n such that there is a unique integer k for which the equation 8/15 = (k - n)/n holds.

Let's simplify the equation:

8/15 = (k - n)/n

Multiply both sides of the equation by 15n to eliminate the denominators:

8n = 15(k - n)

Expand the equation:

8n = 15k - 15n

Rearrange the terms:

23n = 15k

Now, we need to find the largest positive integer n that satisfies this equation. We can start by finding the smallest positive integer k that satisfies this equation.

Let's try n = 1:

23(1) = 15k
23 = 15k

This equation does not have a unique integer solution for k. So, n = 1 is not the largest positive integer that satisfies the equation.

Let's try n = 2:

23(2) = 15k
46 = 15k

This equation does not have a unique integer solution for k. So, n = 2 is not the largest positive integer that satisfies the equation.

Let's try n = 3:

23(3) = 15k
69 = 15k

This equation does not have a unique integer solution for k. So, n = 3 is not the largest positive integer that satisfies the equation.

Let's try n = 4:

23(4) = 15k
92 = 15k

This equation does not have a unique integer solution for k. So, n = 4 is not the largest positive integer that satisfies the equation.

Let's try n = 5:

23(5) = 15k
115 = 15k

This equation does not have a unique integer solution for k. So, n = 5 is not the largest positive integer that satisfies the equation.

Let's try n = 6:

23(6) = 15k
138 = 15k

This equation does not have a unique integer solution for k. So, n = 6 is not the largest positive integer that satisfies the equation.

Let's try n = 7:

23(7) = 15k
161 = 15k

This equation does not have a unique integer solution for k. So, n = 7 is not the largest positive integer that satisfies the equation.

Let's try n = 8:

23(8) = 15k
184 = 15k

This equation does not have a unique integer solution for k. So, n = 8 is not the largest positive integer that satisfies the equation.

Let's try n = 9:

23(9) = 15k
207 = 15k

This equation does not have a unique integer solution for k. So, n = 9 is not the largest positive integer that satisfies the equation.

Let's try n = 10:

23(10) = 15k
230 = 15k

This equation does not have a unique integer solution for k. So, n = 10 is not the largest positive integer that satisfies the equation.

Let's try n = 11:

23(11) = 15k
253 = 15k

This equation does not have a unique integer solution for k. So, n =