A proof of the "necessity" that "a, b, c" be expressed by Euclid's formula for any primitive Pythagorean triple is as follows. All such triples can be written as ("a", "b", "c") where and "a", "b", "c" are coprime. Thus "a", "b", "c" are pairwise coprime (if a prime number divided two of them, it would be forced also to divide the third one). As "a" and "b" are coprime, at least one of them is odd, so we may suppose that "a" is odd, by exchanging, if needed, "a" and "b". This implies that "b" is even and "c" is odd (if "b" were odd, "c" would be even, and "c"2 would be a multiple of 4, while "a"2 + "b"2 would be congruent to 2 modulo 4, as an odd square is congruent to 1 modulo 4).