Question 86
Let $a$, $m$ be positive integers such that $\text{ord}_m(a)$ is odd and for any integers $x$, $y$ such that:
a) $xy \equiv a \pmod{m}$.
b) $\text{ord}_m(x) \le \text{ord}_m(a)$.
c) $\text{ord}_m(y) \le \text{ord}_m(a)$.
We have either $\text{ord}_m(x) \mid \text{ord}_m(a)$ or $\text{ord}_m(y) \mid \text{ord}_m(a)$. Prove that $\text{ord}_m(a)$ has at most one prime divisor.
a) $xy \equiv a \pmod{m}$.
b) $\text{ord}_m(x) \le \text{ord}_m(a)$.
c) $\text{ord}_m(y) \le \text{ord}_m(a)$.
We have either $\text{ord}_m(x) \mid \text{ord}_m(a)$ or $\text{ord}_m(y) \mid \text{ord}_m(a)$. Prove that $\text{ord}_m(a)$ has at most one prime divisor.