# Defining a set of all possible non-congruent integer values

"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)

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