Question

Find the $HCF$ of $65$ and $117$ and express it in the form $65m+117n.$

Answer 1

Given integers are $65$ and $117$ such that $117>65$

Applying division lemma to $65$ and $117$, we get
$117=65\times 1+52$
Since the remainder $52\ne 0$. So, apply the division lemma to the divisor $65$ and the remainder $52$ to get
$65=52\times 1+13$

We consider the new divisor $52$ and the new remainder $13$ and apply division lemma, to get
$52=13\times 4+0$

At this stage the remainder is zero. So, that last divisor or the non-zero remainder at the earlier stage i.e. $13$ is the $HCF$ of $65$ and $117$.

From $\left(ii\right)$, we have
$13=65-52\times 1$
$\Rightarrow 13=65-(117-65\times 1)$
$\Rightarrow 13=65-117+65\times 1$
$\Rightarrow 13=65\times 2+117\times (-1)$
$\Rightarrow 13=65m+117n$, where $m=2$ and $n=-1$.