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: 2019-04-10 21:54:59 | by: *bookofproofs* | references: [1272], [8152]

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

[8152] **Jones G., Jones M.**: “Elementary Number Theory (Undergraduate Series)”, Springer, 1998

[1272] **Landau, Edmund**: “Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie”, S. Hirzel, Leipzig, 1927

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