Why the two equations are equivalent.

Question

In my book ,it says the two equations $x\equiv c\frac{b}{d}\pmod{\frac{m}{d}}$,$ax\equiv b\pmod{m}$ are equivalent where $d=gcd(a,m)$and $c\frac{a}{d}\equiv 1\pmod{\frac{m}{d}}$,why the two equations are equivalent

Answer 1

Assume that $a$ and $m$ are co-prime. Then your question becomes: assuming that $ca\equiv1\pmod m$, why are the equations $x\equiv cb\pmod m$ and $ax\equiv b\pmod m$ equivalent? That's because$$x\equiv cb\pmod m\iff ax\equiv acb\pmod m\iff ax\equiv b\pmod m.$$Can you do the general case now?