Problem
NT-B2-M01-P010 Coprime Exponents
#10
★★★☆☆ Level 3 of 5
Let \(a>1\), and let \(m,n\) be positive integers with \( \gcd(m,n)=1 \). Prove that \( \gcd(a^m-1,a^n-1)=a-1 \).
Use the general fact \( \gcd(a^m-1,a^n-1)=a^{\gcd(m,n)}-1 \), or prove it by the Euclidean algorithm.
Since \( \gcd(m,n)=1 \), the general power identity gives \( \gcd(a^m-1,a^n-1)=a^1-1=a-1 \). Directly, the common divisor is preserved when the exponent pair \((m,n)\) is replaced by \((m-n,n)\), until the Euclidean algorithm reaches exponent \(1\).
This is short, but it reinforces the main power template.