# If the H.C.F of $65$ and $117$ can be written as $65m - 117,$ find the value of $m$?

Last updated date: 17th Mar 2023

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Answer

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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 Euclid’s division Algorithm to get the given form

Euclid’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 Euclid’s division Algorithm to get the given form

Euclid’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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