# Prove that two congruences by modulo are equivalent

Given $a \equiv a'\pmod m$ and $b \equiv b' \pmod m$.

Prove that $ax \equiv b \pmod m$ and $a'x \equiv b' \pmod m$ are equivalent.

I understand I have to show that this modulo congruences have the same set of solutions. But what is solution?

• You do not have to find the solutions to these. Just show that if some $x$ verifies the first, then it verifies the second, and reciprocally, if it verifies the second, then it verifies the first. – Vincent Dec 4 '16 at 15:49

Note $\ a\equiv a',\, b\equiv b'\,\Rightarrow\, ax-b\equiv a'x-b'\,$ by the Congruence Product and Sum Rules.
Therefore we conclude $\,ax-b\equiv 0\iff a'x-b'\equiv 0,\$ i.e. $\ ax\equiv b\iff a'x\equiv b'$
Remark $\$ Alternatively we can note that $\,f(a,b) = ax-b\,$ is a polynomial in $\,a,b\,$ with integer coefficients, therefore $\, a,b\equiv a',b'\,\Rightarrow\, f(a,b)\equiv f(a',b')\,$ by the Polynomial Congruence Rule.