Problem
NT-B2-M01-P011 Neighbouring Powers
#11
★★★☆☆ Level 3 of 5
Prove that \( \gcd(3^n-1,3^n+2)=1 \) for every positive integer \(n\).
The difference of the two numbers is \(3\), but check divisibility by \(3\).
A common divisor divides the difference \((3^n+2)-(3^n-1)=3\). Thus the only possible common prime divisor is \(3\). But \(3^n-1\equiv -1\pmod3\), so \(3\) does not divide the first number. Hence the gcd is \(1\).
A useful check: when a common divisor divides a small number, every prime divisor of that small number must be tested.