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Hint: Here, first we have to find H.C.F of the given numbers then use Euclid's division Algorithm to find the value of m from the above expression.

Let us Euclidâ€™s division Algorithm to get the given form

Euclidâ€™s division Algorithm

Dividend = divisor $ + $quotient $ \times $remainder

Here largest number will be dividend=$117$

And smallest number will the divisor = $65$

Let apply the given algorithm to get above of H.C.F

$

117 = 1 \times 65 + 52 \\

65 = 1 \times 52 + 13 \\

52 = 4 \times 13 + 0 \\

$

Here H.C.F of given numbers is $13$since the remainder is $0$

Let us equate the H.C.F value to the given form

$

\Rightarrow 65m - 117 = 13 \\

\Rightarrow 65m = 13 + 117 \\

\Rightarrow 65m = 130 \\

\Rightarrow m = \dfrac{{130}}{{65}} \\

\Rightarrow m = 2 \\

$

$\therefore m = 2$

NOTE: Always remember that dividend should be the greater number and divisor should be smaller number. To solve the above given problem

Let us Euclidâ€™s division Algorithm to get the given form

Euclidâ€™s division Algorithm

Dividend = divisor $ + $quotient $ \times $remainder

Here largest number will be dividend=$117$

And smallest number will the divisor = $65$

Let apply the given algorithm to get above of H.C.F

$

117 = 1 \times 65 + 52 \\

65 = 1 \times 52 + 13 \\

52 = 4 \times 13 + 0 \\

$

Here H.C.F of given numbers is $13$since the remainder is $0$

Let us equate the H.C.F value to the given form

$

\Rightarrow 65m - 117 = 13 \\

\Rightarrow 65m = 13 + 117 \\

\Rightarrow 65m = 130 \\

\Rightarrow m = \dfrac{{130}}{{65}} \\

\Rightarrow m = 2 \\

$

$\therefore m = 2$

NOTE: Always remember that dividend should be the greater number and divisor should be smaller number. To solve the above given problem

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