Question

If HCF of 65 and 117 is expressible in the form $65n - 117$, then find the value of $n$.

Answer 1

Hint: We will first write the numbers 65 and 17 as the product of their prime factors. Then, we will find the highest common factor of 65 and 117. Equate the HCF of 65 and 117 to $65n - 117$. Further, solve the equation to find the value of \[n\].

Complete step-by-step answer:
We will first calculate the HCF of 65 and 117.
We will factorise 65 and 117 and will write the numbers as the product of their prime factors.

5	65
13	13
	1

$65 = 5 \times 13$

3	117
3	39
13	13
	1

$117 = 3 \times 3 \times 13$
Now, the highest common factor of 65 and 117 is 13.
According to the question, $65n - 117 = 13$
Add 117 to both sides,
$
  65n = 13 + 117 \\
   \Rightarrow 65n = 130 \\
$
Divide both sides by 65
$n = 2$
Hence, the value of $n$ is 2.

Note: Highest common factor of two numbers can also be calculated using division method and factor tree-method. Here, the correct formation of the equation depends on the HCF of the two numbers. HCF of the numbers is the highest common number that divides the numbers completely.