## abc-hits

### Problem 127

Published on Friday, 1st September 2006, 06:00 pm; Solved by 3784; Difficulty rating: 50%The radical of *n*, rad(*n*), is the product of distinct prime factors of *n*. For example, 504 = 2^{3} × 3^{2} × 7, so rad(504) = 2 × 3 × 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*) = 1 *a*<*b**a*+*b*=*c*- rad(
*abc*) <*c*

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

- GCD(5, 27) = GCD(5, 32) = GCD(27, 32) = 1
- 5 < 27
- 5 + 27 = 32
- rad(4320) = 30 < 32

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

Find ∑*c* for *c* < 120000.