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

Hint: Apply prime factorization and get the factors of 65 and 117. 13 is the common factor for 65 and 117. We know the property that the HCF of two numbers is the number that is common in the factors of both numbers. Now, compare 13 with \[65m-117\] and calculate the value of \[m\] .

Complete step by step answer:
According to the question, we are given two numbers 65, 117 and also their HCF is given using an expression in terms of \[m\] and we are asked to find the value of \[m\] .
The first number = 65 …………………………………..(1)
The second number = 117 ……………………………………….(2)
The HCF of 65 and 117 = \[65m-117\] ………………………………………(3)
First of all, we have to apply prime factorisation in order to get the factors of 65 and 117.
Now, on applying prime factorisation, we get
\[65=5\times 13\] ……………………………………(4)
\[117=3\times 3\times 13\] …………………………………………..(5)
Now, from equation (4) and equation (5), we can observe that 13 is the common factor for both numbers ……………………………………(6)
We also know the property that the HCF of two numbers is the number that is common in the factors of both numbers ……………………………………..(7)
 From equation (6) and equation (7), we can say that 13 is the HCF of 65 and 117.
So, the HCF of 65 and 117 is 13 …………………………………..(8)
Now, from equation (3) and equation (8), we get
\[\begin{align}
  & \Rightarrow 65m-117=13 \\
 & \Rightarrow 65m=117+13 \\
 & \Rightarrow 65m=130 \\
 & \Rightarrow m=\frac{130}{5} \\
\end{align}\]
\[\Rightarrow m=2\] …………………………………(9)
Therefore, the value of \[m\] is 2.
Hence, the correct option is (B).

Note:
 Here, one might approach this question by using the traditional method of calculating the HCF of two numbers. Using the traditional method will take time and might lead to some calculation mistake. Therefore, use the property that the HCF of two numbers is the number that is common in the factors of both numbers. Using this property will reduce the complexity and save time.