Problem 127

abc-hits

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.

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