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Proof

(related to Proposition: Multiplication of Congruences with a Positive Factor)

By hypothesis, $a,b$ are integers and $c > 0, m > 0$ are positive integers.

"$\Rightarrow$"

• Since $c > 0,$ the divisibility law no. 8 implies $m\mid (a-b)\Rightarrow mc\mid (a-b)c.$
• By the distributivity law of integers, $m\mid (a-b)\Rightarrow mc\mid (ac-bc).$
• This means, by definition of congruences, $a(m)\equiv b(m)\Rightarrow (ac)(mc)\equiv(bc)(mc).$

"$\Leftarrow$"

• The divisibility law no. 7 implies (for all $c\in\mathbb Z,$ not only $c > 0$) that $mc\mid (a-b)c\Rightarrow m\mid (a-b).$
• By the distributivity law of integers, $mc\mid (ac-bc)\Rightarrow m\mid (a-b).$
• This means, by definition of congruences, $(ac)(mc)\equiv(bc)(mc)\Rightarrow a(m)\equiv b(m).$
  ∎

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References

Bibliography

1. Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927