Prove $ax \equiv b \mod m \iff b \equiv 0 \mod gcd(m, a)$.

Question

Prove $ax \equiv b \mod m \iff b \equiv 0 \mod gcd(m, a)$.

I have been stuck for quite some time, but here is my proof so far:

Let $gcd(m, a) = d$. Then $b = dk$.

$ax \equiv b \mod m \implies ax - qm = b$.

A solution exists if and only if $x \in \mathbb{Z}$. So solving for $x$, we find that $x = \frac{b+qm}{a}$. Because $d | m \land d | a$, it can be said that $x = d(\frac{k + qj}{i})$ where $m = dj$ and $a = di$.

I have a feeling that I can show that $x$ is an integer by doing something like this, but I'm not quite sure how.

Answer 1

I'm assuming you mean that there exists a solution to $ax \equiv b \mod m$ if and only if $b \equiv 0 \mod \gcd(m,a)$.

Let $d = \gcd(m,a)$.

Suppose $ax \equiv b \mod m$ for some integer $x$. Then there is an integer $k$ such that $ax-b=km \, \Longrightarrow \, b = ax -km$. Because $d$ divides both $a$ and $m$ it also divides $ax - km=b$ which is to say that $b \equiv 0 \mod d$.

Conversely, suppose $b \equiv 0 \mod d$. Then $b = rd$ for some integer $r$. Express $d = pm+qa$ as a integer combination of $m$ and $a$, which is possible by virtue of the euclidean algorithm. It follows that $b = rd = rpm + rqa \equiv (rq)a \mod m$. letting $x = rq$ we rewrite this as $ax \equiv b \mod m$.