data/problems/127.yml
---
:id: 127
:name: abc-hits
:url: https://projecteuler.net/problem=127
:content: |+
The radical of _n_, rad(_n_), is the product of distinct prime factors of _n_. For example, 504 = 2<sup>3</sup> × 3<sup>2</sup> × 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:
1. GCD(_a,_ _b_) = GCD(_a_, _c_) = GCD(_b_, _c_) = 1
2. _a_ \< _b_
3. _a_ + _b_ = _c_
4. rad(_abc_) \< _c_
For example, (5, 27, 32) is an abc-hit, because:
1. GCD(5, 27) = GCD(5, 32) = GCD(27, 32) = 1
2. 5 \< 27
3. 5 + 27 = 32
4. 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.