Proof
(related to Proposition: Congruence Classes)
- Let $m > 0$ be a positive integer and let $a\equiv b(m)$ (two integers $a,b$ being congruent modulo $m$).
- We show three properties, applying the definitions of congruence and divisors:
- Reflexivity: $m\mid (a-a)=0$, therefore $a\equiv a(m).$
- Symmetry: $m\mid (a-b)\Leftrightarrow m\mid (b-a),$ therefore $a\equiv b(m)\Leftrightarrow b\equiv a(m).$
- Transitivity:
* If $a\equiv b(m)$ and $b\equiv c(m)$, then $m\mid(a-b)$ and $m\mid(b-c).$
* But then $m\mid (a-b)+(b-c)=(a-c)$
* It follows $a\equiv c(m).$
- Altogether, it follows that the relation $[\equiv(m)\subset \mathbb Z\times\mathbb Z$ is an equivalence relation.
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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
