The following proposition generalizes the previous proposition, which proves to be a special case of the following one 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: 2019-06-22 15:20:11 | by: bookofproofs | references: [1272], [8152]

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