Problem

Source: 2008 Indonesia TST stage 2 test 3 p3

Tags: GCD, greatest common divisor, number theory



Let $n$ be an arbitrary positive integer. (a) For every positive integers $a$ and $b$, show that $gcd(n^a + 1, n^b + 1) \le n^{gcd(a,b)} + 1$. (b) Show that there exist infinitely many composite pairs ($a, b)$, such that each of them is not a multiply of the other number and equality holds in (a).