Question

If the HCF of 65 and 117 is expressible in the form \[65n - 117\]. Find the value of n?

Answer 1

Hint:
In this solution, we will solve the question by using the prime factorisation method where we will expand the numbers in the form of its prime factor multiplication whose HCF is required. After finding the HCF we will equate the value to the equation given and find the value of n.

Complete step by step solution:
Given, HCF of numbers 65 and 117 = \[65n - 117\]………………..(1) where HCF stands for highest common factor.
By prime factorisation method,
We will now represent the number 65 as the product of its prime factors as shown below
$65 = 5 \times 13$
Here, prime factors of the number 65 are 5 and 13.
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 which is occurring twice 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 the case of the two numbers 65 and 117, we have only one common prime factor which is 13.
So, HCF of two numbers 65 and 117 = 13
According to equation (1), above equation becomes
\[ \Rightarrow 65n - 117 = 13\]
Now adding both the sides by 117, we get:
\[ \Rightarrow 65n = 13 + 117\]
\[ \Rightarrow 65n = 130\]
Now we divide both the sides by 65 and we get:
\[ \Rightarrow n = \dfrac{{130}}{{65}} = 13\]

Therefore, the value of n is 2.

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.