# Modular Multiplicative Inverse when $A \gt M$

You are given two positive integers $A$ and $M$ where $\gcd(A, M) = 1$. You have to determine whether there exists at least one integer $X$ such that $AX = 1 \pmod M$.

Okay, this is actually finding the modular multiplicative inverse exists. But here $A$ can be greater than $M$. $A$ and $M$ are $\geq 1$. So is there any case that I need to consider when $A \gt M$?

No consideration is required. Because there exist $A_1<M$ so that
$A \equiv A_1 \mod{M}$
$A_1$ is relatively prime to M.
$(A_1,M)=(Mk+A,M)=(A,M)=1$
Therefore $A_1$ has an inverse (mod M) and So has $A$.