Problem

Source: Iranian third number theory finals problem 3

Tags: number theory



Let $a,m$ be positive integers such that $Ord_m (a)$ is odd and for any integers $x,y$ so that 1.$xy \equiv a \pmod m$ 2.$Ord_m(x) \le Ord_m(a)$ 3.$Ord_m(y) \le Ord_m(a)$ We have either $Ord_m(x)|Ord_m(a)$ or $Ord_m(y)|Ord_m(a)$.prove that $Ord_m(a)$ contains at most one prime factor.