Proof
(related to Proposition: Congruence Modulo a Divisor)
- By hypothesis, $a,b$ are integers, $n > 0, m > 0$ are positive integers, and $m\mid n.$
- By definition of divisor, there is an integer $c > 0$ such that $n=cm.$
- By hypothesis $a(n)\equiv b(n),$ which means $a(cm)\equiv b(cm).$
- By the definition of congruence, $cm\mid (a-b).$
- By the divisibility law no. 3, since $m\mid cm$, it follows $m\mid (a-b).$
- Therefore, $a(m)\equiv b(m).$
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References
Bibliography
- Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927
