Question 4
Write $\gcd(a,b)$ for the greatest common divisor of natural numbers $a,b$. Prove that, for natural numbers $a,b,m$, a necessary and sufficient condition for the existence of a natural number $n$ such that $(a^n-1)b$ is divisible by $m$ is
$$\gcd(ab,m)=\gcd(b,m).$$
$$\gcd(ab,m)=\gcd(b,m).$$