Question

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

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