Unlike addition, subtraction and multiplication of congruences, there is no general way to divide congruences. However, in some cases, we can simplify a given congruence $ac\equiv bc\mod m$ by canceling out the factor $c$. The following proposition shows that this is only possible if $c$ and the module $m$ are coprime.

**Proposition**: Cancellation of Congruences With Factor Co-Prime To Module

Let the $a,b,c$ be integers, and $m > 1$ be a positive integer and let $c\perp m$ be co-prime. Then, from the equaility of the congruences $$(ac)(m)\equiv (bc)(m)$$ it follows that $$a(m)\equiv b(m).$$

| | | | | created: 2019-04-13 16:36:48 | modified: 2019-04-19 08:55:49 | by: *bookofproofs* | references: [1272], [8152]

## 1.**Proof**: *(related to "Cancellation of Congruences With Factor Co-Prime To Module")*

[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

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