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

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Solution

Step $1$ : Find the HCF of $65$ and $117$ .
Using Euclid’s division algorithm,
$117 = 65 \times 1 + 52 \Rightarrow 65 = 52 \times 1 + 13 \Rightarrow 52 = 13 \times 4 + 0$
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:
$65 m - 117 = 13 \Rightarrow 65 m = 130 \Rightarrow m = \frac{130}{65} \Rightarrow m = 2$
Hence, the value of $m$ is $2$ .

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