Given:
- N and x are positive integers
- NN = 2160
- N^2 - 2N is an integral multiple of 2x

To find:
The largest possible value of x

Solution:

Step 1: Finding the value of N
Given that NN = 2160, we need to find the value of N.

Taking the square root on both sides, we get:
N = √2160

Using prime factorization, we can write 2160 as a product of its prime factors:
2160 = 2^4 * 3^3 * 5^1

Taking the square root of 2160 gives us:
N = √(2^4 * 3^3 * 5^1)
= (2^2 * 3^1 * 5^(1/2))

Simplifying further, we get:
N = 2^2 * 3 * 5^(1/2)
= 4 * 3 * √5
= 12√5

So, N = 12√5.

Step 2: Finding the value of N^2 - 2N
We are given that N^2 - 2N is an integral multiple of 2x.

Substituting the value of N, we have:
(12√5)^2 - 2(12√5)

Simplifying further, we get:
144 * 5 - 24√5
720 - 24√5

Step 3: Finding the largest possible value of x
We need to find the largest possible value of x such that 720 - 24√5 is an integral multiple of 2x.

Let's assume 720 - 24√5 = 2kx, where k is an integer.

Rearranging the equation, we get:
720 - 2kx = 24√5

Squaring both sides, we get:
(720 - 2kx)^2 = (24√5)^2
518,400 - 2 * 720 * 2kx + (2kx)^2 = 5 * 24^2

Simplifying further, we get:
518,400 - 2 * 720 * 2kx + (2kx)^2 = 5 * 576
518,400 - 2,073,600kx + 4k^2x^2 = 2,880

Rearranging the equation, we have:
4k^2x^2 - 2,073,600kx + 515,520 = 0

Now, we need to find the values of k and x that satisfy this quadratic equation.

Step 4: Finding the largest possible value of x
Since we are looking for positive integer solutions, the discriminant of the quadratic equation must be a perfect square.

The discriminant, Δ = b^2 - 4ac, where a = 4k^2, b = -2,073,600k, and c = 515,520.

Substituting the values, we get:
Δ = (-2,073