Theory:

Let $$a$$ and $$b$$ be two integers, where $$a$$ and $$b$$ are positive integers. Then, by Euclid's division lemma, we know that $$a = bq + r$$ where $$0 \leq r < b$$ and $$q$$ is an integer. Now, let us apply the congruence modulo and for the given Euclid's division lemma.

Thus, from the given, we can say that $$a$$ is congruent to $$r$$ modulo $$b$$, for some integer $$q$$.

That is, $$a = bq + r$$

$$a - r = br$$

$$a - r \equiv 0 (mod \ b)$$

$$a \equiv r (mod \ b)$$

Therefore, using Euclid's division lemma, the equation $$a = bq + r$$ can be written as $$a \equiv r (mod \ b)$$.

Important!
Two integers, $$a$$ and $$b$$, are said to be congruent to modulo $$m$$, if they both receive the same remainder when divided by $$m$$.