Question

If the H.C.F. of $65$ and $117$ is expressible in form $65m–117$, calculate the value of $m$?

Answer 1

Step $1$ : Find the HCF of $65$ and $117$.

Using Euclid’s division algorithm,

$$\begin{gathered} 117=65\times 1+52 \\ \Rightarrow 65=52\times 1+13 \\ \Rightarrow 52=13\times 4+0 \end{gathered}$$

Since the remainder has come to zero so the H.C.F. of given numbers is $13$.

Step $2$ : Find the required value:

Given that H.C.F. is expressed as following:

$$\begin{gathered} 65m–117=13 \\ \Rightarrow 65m=130 \\ \Rightarrow m=\frac{130}{65} \\ \Rightarrow m=2 \end{gathered}$$

Hence, the value of $m$ is $2$.