If x and y are positive integers, L: LCM of x and y, G: GCD of x and y, then solve the following equations for x and y values:
1) x^y=L^G, L>x.
2) x²+y² = L²+G², L>x>G.

Let a = x/G, b = y/G

Then L = Gab.

Substituting into equation 1 yields

(Ga)^(Gb) = (Gab)^G

Simplifying,

(Ga)^b = Gab

(Ga)^(b-1) = b

The only integral solutions are is b =2, Ga = 2
      and b = 1.  But if b = 1, then Ga = x = L, which is ruled out by the problem.

If b = 2, then

G = 1, a = 2 does not lead to a solution, because this leads to
x = 2, y = 2, but the GCD of 2 and 2 is 2, not 1

So the only solution is G = 2, a = 1, b = 2
Then x = 2, y = 4, L = 4, G = 2.

Posted by Steve Herman on 2007-12-22 15:18:13