Question

If the HCF of 65 and 117 is expressible in the form 65m-117, then what is the value of m?
$
  {\text{A}}{\text{. 4}} \\
  {\text{B}}{\text{. 2}} \\
  {\text{C}}{\text{. 1}} \\
  {\text{D}}{\text{. 3}} \\
 $

Answer 1

Hint: Here, we will proceed by using the prime factorisation method in which we will represent the numbers whose HCF is required (in this case these numbers are 65 and 117) in a form of multiplication of its prime factors.

Complete step-by-step answer:
Given, HCF of numbers 65 and 117 = 65m-117 $ \to (1)$ where HCF stands for highest common factor.

By prime factorisation method,

Let us represent the number 65 as the product of its prime factors as given below

$65 = 5 \times 13$

Here, both 5 and 13 are the prime factors of the number 65.

Similarly, let us represent the number 117 as the product of its prime factors as given below

$117 = 3 \times 3 \times 13$

Here, both 3 and 13 are the prime factors of the number 117.

The HCF of any two numbers can be obtained by multiplying all the common prime factors

between the two numbers as many times as they are occurring commonly.

In case of the numbers 65 and 117, there is only one common prime factor which is 13

So, HCF of numbers 65 and 117 = 13

According to equation (1), above equation becomes

$

  65m - 117 = 13 \\

   \Rightarrow 65m = 13 + 117 \\

   \Rightarrow 65m = 130 \\

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

   \Rightarrow m = 2 \\

 $

Therefore, the value of m is 2.

Hence, option B is correct.

Note: In this particular problem, there is only one prime factor common between the numbers whose HCF is required. If we are having more than one prime factor common between the numbers in that case the product of the common prime factors is taken and also if the same prime factor is common multiple times in that case that prime factor is multiplied with itself as many times as they are common.