The following proposition demonstrates the division with quotient and remainder we have introduced already provides an equivalent possibility to define congruences.

Proposition: Congruences and Division with Quotient and Remainder 

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
$$a\equiv b(m)\Longleftrightarrow r_a=r_b.$$

| | | | | created: 2019-04-10 21:28:19 | modified: 2019-06-20 08:16:56 | by: bookofproofs | references: [1272], [8152]

1.Proof: (related to "Congruences and Division with Quotient and Remainder")