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I have the following statement to prove, for homework:

If $r ∈ \mathbb{z}$ is a solution to $ax ≡ b ($ mod $m)$, then $r + mk, k ∈ Z$, are also solutions to $ax ≡ b ($ mod $ m)$.

What does "$r ∈ \mathbb{z}$ is a solution..." mean? I assume it means $r=b$ since $b$ is usually where the modulo's solution is, but by symmetry the equation can be written as $b ≡ ax ($ mod $m)$, and now $ax$ would be the solution. Any insight on this problem is appreciated. Thank you

mathjohnn
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    i expect that $a,b,m$ are given and that you are trying to solve for $x$. Saying that an integer $r$ is a solution then means that $a\times r\equiv b\pmod m$. – lulu Dec 05 '20 at 18:54
  • Thank you! I’ll try to solve with that. – mathjohnn Dec 05 '20 at 18:56

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