not really elegant (it is more intuitive).anyways, my approach-

5^x-5^y=2^y/2*5^x/5

10(5^x-5^y)=2^y*5^x

I picked answer choice E 12,since it appears to be the easiest one.

12=3*4

let x=4 y=3

10(5^4-5^3)=2^3*5^4
5^3*2*5(5-1)=2^3*5^4
5^3*2^3*5=2^3*5^3*5

If x and y are positive integers and 5^x - 5^y = 2^(y-1) * 5^(x-1), what is xy?

A)48 B)36 C)24 D)18 E)12

Since 5^(x-1) is a factor of the righthand side, 5^(x-1) must be a factor of the lefthand side.

(5^x - 5^y) / 5^(x-1) = 2^(y-1) * 5^(x-1) / 5^(x-1) Divide each side by 5^(x-1).

5^(x-x+1) - 5^(y-x+1) = 2^(y-1) Subtract the exponents on the lefthand side and simplify the righthand side.

5 - 5^(y-x+1) = 2^(y-1).

Since the righthand side is a positive integer, so must be the lefthand side.
Thus, we know that 5^(y-x+1) = 5� = 1.
Otherwise, the lefthand side will not be a positive integer.

Substituting 5^(y-x+1)=1 into 5 - 5^(y-x+1) = 2^(y-1), we get:
5-1 = 2^(y-1)
4 = 2^(y-1)
y=3.

Since y-x+1=0, we get:
3-x+1 = 0
x=4.

Thus, xy = 4*3 = 12.

The correct answer is E.