Problem 127 of Project Euler is a really awesome number theoretic problem using concepts of GCD and radicals. The problem reads

The radical ofn, rad(n), is the product of distinct prime factors ofn. For example, 504 = 2^{3}x 3^{2}x 7, so rad(504) = 2 x 3 x 7 = 42.

We shall define the triplet of positive integers (a,b,c) to be an abc-hit if:

GCD(a,b) = GCD(a,c) = GCD(b,c) = 1a<ba+b=crad(abc) <c

For example, (5, 27, 32) is an abc-hit, because:

GCD(5, 27) = GCD(5, 32) = GCD(27, 32) = 15 < 275 + 27 = 32rad(4320) = 30 < 32

It turns out that abc-hits are quite rare and there are only thirty-one abc-hits forc< 1000, with∑c= 12523.

Find∑cforc< 120000.