Question

If$\left(a,c\right)=1and\left(b,c\right)=1$ Prove that $\left(ab,c\right)=1$.

Short answer

It is proved that $\left(ab,c\right)=1$.

Step-by-step solution

Definition of Greatest Common Divisor: 

Let $d$ be the largest integer that divides the integers $aandb$. Then the integer can be designated as:

$d=\left(a,b\right)$

Where, $a,b\ne 0$.

Proof using Definition of GCD:

Let $\left(a,b\right)|d$, then $d|c$.

Then using the definition of GCD, we have:

$$\begin{gathered} \left(a,d\right)\left(a,c\right)=1 \\ \left(a,d\right)=1 \end{gathered}$$

Similarly, we get:

$\left(b,d\right)=1$

Also, we already know that $d|ab$and $\left(a,d\right)=1$. This implies,

$d|b$

So, from these expressions, we get:

$d=\left(b,d\right)=1$

Hence, it is proved that $\left(ab,c\right)=1$.