how to solve $min\{x \in \mathbb N_0 \quad |x \cdot 714\quad mod \quad 1972 \quad = \quad 1292 \quad mod \quad 1972 \} $ (modulo equation) Question: How can I solve: $min\{x \in \mathbb N_0 \quad |x \cdot 714 \equiv 1292 \mod 1972 \} $ ?
I only know about:  
$x \cdot a \equiv _m b \Rightarrow m|x \cdot a - b$
different way of notation:
$min\{x \in \mathbb N_0 \quad | x \cdot 714 \equiv_{1972} 1292\}$ 
How to go on?
I appreciate every hint. 

Let's say you want to solve $ax \equiv b \mod{m}$. If $gcd(a,m)=1$, then there is a unique solution. (You should be able to get this solution)
Now if $gcd(a,m) = d$, then solution exists iff $b$ divides $d$. You should try to reduce this to a equation between $\frac{a}{d}, \frac{b}{d} $ and $\frac{m}{d}$. From this solution , you will get mutiple solutions for the initial equation. Can you easily find the min of that?
Further hint for the reduction - Solving $ax \equiv b \mod{m}$ is same as solving $ax = b + my$.
