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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 $a-b.$

Table of Contents

Explanations: 1

1. Proposition: Congruence Classes
2. Proposition: Congruences and Division with Quotient and Remainder
3. Proposition: Connection between Quotient, Remainder, Modulo and Floor Function
4. Lemma: Coprimality and Congruence Classes
5. Proposition: Multiplication of Congruences with a Positive Factor
6. Proposition: Congruence Modulo a Divisor
7. 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

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