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A generalization of the previous proposition about the cancellation of congruences is the following proposition, in which the previous proposition is a special case with $\gcd(c,m)=1.$

## Proposition: Cancellation of Congruences with General Factor

Let the $a,b,c$ be integers, and $m > 1$ be a positive integer and let $d=\gcd(c,m)$ be the greatest common divisor of $c$ and $m$. Then, from the equaility of the congruences $$(ac)(m)\equiv (bc)(m)$$ it follows that $$a\left(\frac md\right)\equiv b\left(\frac md\right).$$

### Example

$3\cdot 2(8)\equiv 7\cdot 2(8)\Longrightarrow 3(4)\equiv 7(4).$

| | | | | created: 2019-04-13 17:21:46 | modified: 2019-04-19 08:56:36 | by: bookofproofs | references: [1272], [8152]

## 1.Proof: (related to "Cancellation of Congruences with General Factor")

### This work is a derivative of:

(none)

[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