Problem

Source: Indian Postal Coaching 2005

Tags: number theory, greatest common divisor, algorithm, Euclidean algorithm, number theory solved



Let $m,n$ be natural numbers and let $d = gcd(m,n)$. Let $x = 2^{m} -1$ and $y= 2^n +1$ (a) If $\frac{m}{d}$ is odd, prove that $gcd(x,y) = 1$ (b) If $\frac{m}{d}$ is even, Find $gcd(x,y)$