Taken from here (last page).
Let $t\in\mathbb Z$ and let $0\le j\le g-1$ such that $j\equiv t \pmod g$.
Then,$$t\cdot \frac {m}{g}\equiv j\cdot \frac{m}{g}\pmod m$$
($g = \gcd (m,a)$ if it matters)
Why is that?
Also, if I'm trying to divide by $m$ I get:
$$\frac{t}{g} \equiv \frac{j}{g} \pmod {1} \implies \frac{t}{g} = \frac{j}{g} \implies j=t$$
but it doesn't have to be the case at all that $j=t$.
I'd be glad if you could help me arranging my thoughts around this.
Thanks!
By your assumption, $(t-j)/g$ is an integer $c$. Therefore $$t\frac{m}{g}-j\frac{m}{g}=cm.$$ Is this divisible by $m$?