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

Proofs: 1

Explanations: 1
Proofs: 2 3 4
Propositions: 5

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  1. Landau, Edmund: "Vorlesungen ├╝ber Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927
  2. Jones G., Jones M.: "Elementary Number Theory (Undergraduate Series)", Springer, 1998