Question

If the HCF of 65 and 117 can be expressed as (65m – 117), find the value of m.

Answer 1

Hint: First, find the prime factorisation of 65. Next, find the prime factorisation of 117. Using these two, find the HCF of 65 and 117. Equate this HCF to the given expression: (65m – 117). Solve the equation for m. The resulting value is the final answer.

Complete step-by-step solution -

In this question, we are given that the HCF of 65 and 117 can be expressed as (65m – 117).
We need to find the value of m.
To solve this question, we will first find the HCF of 65 and 117.
For this, we will do the prime factorisation of 65 and 117:
 5 |65
13|13
     |1
So, prime factorisation of 65 = 5 $\times $ 13
Similarly, we will find the prime factorisation of 117:
3 |117
3 |39
3 |13
   |1
So, prime factorisation of 65 = 3 $\times $ 3 $\times $ 13
So, from the above prime factorisations, we see that only 13 is common in both of them.
So, the HCF of 65 and 117 is 13.
Now, we are given that the HCF of 65 and 117 can be expressed as (65m – 117).
So, 65m – 117 = 13
65m = 117 + 13
65m = 130
m = 2
Hence, the value of m is 2.
This is our final answer.

Note: In this question, it is very important to know how to find the HCF of two numbers. In mathematics, the highest common factor of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers.