"Say Peter has discovered that $82$ and $723$ are coprime.

He now believes that the equation below has a solution for all possible integer values of $q$.

$$82p ≡ q\pmod {723},$$

where $p∈$ $\Bbb Z$

Using appropriate and precise mathematical language, define $Q$, the set of all possible non-congruent integer values for $q$.

State, with reason, why Peter is correct regarding the given equation."

With regards to solving this question, I have only found the coefficients of $x$ and $y$ in Bezout's Identity, but am unsure as where to go from there (or if I'm even meant to be there in the first place)

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    $\begingroup$ Can you solve it for $q=1$? $\endgroup$ – Lord Shark the Unknown Nov 10 '18 at 15:28
  • $\begingroup$ Because $82$ and $723$ are coprime, there exists integer $x,y$ satisfying $82x+723y=1$ by Euclidean algorithm. $\endgroup$ – Doyun Nam Nov 10 '18 at 15:42

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