The following proposition demonstrates the division with quotient and remainder we have introduced already provides an equivalent possibility to define congruences.
Let $m > 0$ be the positive integer being the divisor in the division with quotient and remainder of two integers $a,b\in\mathbb Z:$
$$\begin{array}{rcll} a&=&q_am+r_a&0\le r_a < m,\\ b&=&q_bm+r_b&0\le r_b < m. \end{array}$$
Then $a$ is congruent to $b$ if and only if they have the same remainder, formally $$a\equiv b(m)\Longleftrightarrow r_a=r_b.$$
Proofs: 1
Explanations: 1
Proofs: 2 3 4
Propositions: 5