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 $m\mid ab,$ i.e. if $m$ is a divisor of the difference $ab.$
Table of Contents
Explanations: 1
 Proposition: Congruence Classes
 Proposition: Congruences and Division with Quotient and Remainder
 Proposition: Connection between Quotient, Remainder, Modulo and Floor Function
 Lemma: Coprimality and Congruence Classes
 Proposition: Multiplication of Congruences with a Positive Factor
 Proposition: Congruence Modulo a Divisor
 Definition: Modulo Operation for Real Numbers
Mentioned in:
Definitions: 1 2 3
Explanations: 4
Lemmas: 5
Problems: 6 7
Proofs: 8 9 10 11 12 13 14 15 16 17 18 19 20
Propositions: 21 22 23 24 25
Sections: 26
Solutions: 27 28
Theorems: 29 30 31
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References
Bibliography
 Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927
 Jones G., Jones M.: "Elementary Number Theory (Undergraduate Series)", Springer, 1998