Question

Prove that$\left(\left(a,b\right),c\right)=\left(\left(a,b\right),c\right)$
.

Short answer

Hence proved, $\left(\left(a,b\right),c\right)=\left(a,\left(b,c\right)\right)$.

Step-by-step solution

Definition of Greatest Common Divisor:

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

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

Where, $a,b\ne 0$.

Proof using Definition of GCD:

Let $d|\left(b,c\right)$. Then by using the definition, we have:

$d|b\hspace{0.17em}andd|c$.

Also, if $d|a$, then$d$ will be a common divisor of $aandb$. Therefore,

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

So, $d|\left(\left(a,b\right),c\right)$

Similarly, we will have:

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

Hence, it is proved that $\left(\left(a,b\right),c\right)=\left(a,\left(b,c\right)\right)$.