NT-B1-M02-P013 Sum and Product

If a prime \(p\) divides \(ab\), then it divides \(a\) or \(b\).

Suppose a prime \(p\) divides both \(a+b\) and \(ab\). From \(p\mid ab\), we get \(p\mid a\) or \(p\mid b\). If \(p\mid a\), then from \(p\mid a+b\), we get \(p\mid b\), contradicting \(\gcd(a,b)=1\). The case \(p\mid b\) is similar. Thus there are no common prime divisors, and the GCD is \(1\).

Uses Euclid's lemma to prove coprimality.