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Proposition: Congruence Classes

Let $m> 0$ be a positive integer. The relation $”\equiv(m)”$ of being congruent modulo $m$ is an equivalence relation $\equiv(m)\subset\mathbb Z\times \mathbb Z,$ defined on the set of all integers $\mathbb Z.$ In particular, every element $[a]$ of the quotient set, written $a(m)$, is called the congruence class modulo $m$. The quotient set $$\mathbb Z_m:=\mathbb Z/_{\equiv(m)}=\{0(m),1(m),\ldots,(m-1)(m)\}$$ contains the $m$ congruence classes modulo $m.$

| | | | | created: 2019-04-10 21:08:18 | modified: 2019-07-28 09:29:28 | by: bookofproofs | references: [1272], [8152]

1.Proof: (related to "Congruence Classes")

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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