Question

If $c|ab$ and $\left(c,a\right)=d$, prove that$c|db$.

Short answer

Hence it is proved that if $c|ab$and $\left(c,a\right)=d$, then $c|db$.

Step-by-step solution

Linear Combination:

Let d be the largest integer that divides the integers a and b. Then there exist integers u and v such that:

$d=au+bv$

Where$a,b\ne 0$

Proof using definition:

Now, we have: $d=\left(c,a\right)$, therefore using the definition:

$d=cu+av$

role="math" localid="1646108060396" $db=cbu+abv$

Since $c|ab$, therefore,

role="math" localid="1646108149669" $ab=cw.........\left\{w\in Z\right\}$

So, we have:

role="math" localid="1646108518856" $$\begin{gathered} db=cbu+abv \\ =cbu+cwv \\ =c\left(bu+wv\right) \end{gathered}$$

Thus, $c|db$

Hence, it is proved that if $c|ab$and $\left(c,a\right)=d$, then $c|db$.