Problem
NT-B1-M02-P017 Powers of Coprime Numbers
#17
★★★☆☆ Level 3 of 5
Prove: if \(\gcd(a,b)=1\), then \(\gcd(a^m,b^n)=1\) for all positive \(m,n\).
Compare the prime divisors of \(a\) and \(b\).
If a prime \(p\) divides \(a^m\), then \(p\mid a\). If the same \(p\) divided \(b^n\), then \(p\mid b\), contradicting \(\gcd(a,b)=1\). Thus \(a^m\) and \(b^n\) have no common prime divisors, and their GCD is \(1\).
Reinforces transferring coprimality to powers.