Question

If the HCF of 65 and 117 is expressible in the form of 65m - 117, then the value of m is:

• A. 4
• B. 2
• C. 1
• D. 3

Answer 1

The correct option is B 2

Let the HCF of 65 and 117 be x.

Here, 65 = 13 $\times$ 5
117 = 13 $\times$ 3 $\times$ 3
$\Rightarrow$ LCM of 65 and 117 = 585

Now, product of 65 and 117 = HCF(65, 117) $\times$ LCM(65, 117)
$\Rightarrow$ HCF(65, 117) $\times$ 585 = 65 $\times$ 117
$\Rightarrow$ HCF(65, 117) = $\frac{117}{9}$
$\Rightarrow$ HCF(65, 117) = 13

Now, HCF(65, 117) = 13 = 65m - 117
$\Rightarrow$ m = $\frac{130}{65}=2$

Alternate Solution
By prime factorisation ;
117 = 3 $\times$ 3 $\times$ 13
65 = 5 $\times$ 13

Hence HCF of 117 and 65 is 13
HCF(65, 117) = 13 = 65m - 117
Hence m = 2