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

Step 1 : Find the HCF of given numbers

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 is $52\ne 0$. 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 $117$.

Step 2 : Express the HCF in form of given expression

Given expression: $65m+117n$
We can write the HCF of $65$ and $117$as,
$$\begin{gathered} 13=65-52\times 1 \\ \Rightarrow 13=65-(117-65\times 1) \\ \Rightarrow 13=65-117+65\times 1 \\ \Rightarrow 13=65\times 2+117\times (-1) \end{gathered}$$
$\Rightarrow 13=65m+117n$, where $m=2$ and $n=-1$.

Hence, the HCF of $65$ and $117$ is $13$and it can be expressed as $13=65m+117n$, where $m=2$ and $n=-1$.