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Definition: Congruent, Residue

Let $m > 0$ be a positive integer. $a$ is called congruent to $b$ (or $a$ is called a residue of $b$) modulo $m$, written $$a\equiv b \mod m$$
or shorter
$$a\equiv b (m),$$
if $m\mid a-b,$ i.e. if $m$ is a divisor of the difference $a-b.$

| | | | | created: 2019-04-10 20:21:58 | modified: 2019-07-28 09:28:53 | by: bookofproofs | references: [1272], [8152]

1.Proposition: Congruence Classes

2.Proposition: Congruences and Division with Quotient and Remainder

3.Explanation: Explanation of Congruence Classes

4.Proposition: Connection between Quotient, Remainder, Modulo and Floor Function

5.Lemma: Coprimality and Congruence Classes

6.Proposition: Multiplication of Congruences with a Positive Factor

7.Proposition: Congruence Modulo a Divisor

8.Definition: Modulo Operation for Real Numbers

Edit or AddNotationAxiomatic Method

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Bibliography (further reading)

[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