Question

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

Answer 1

The given numbers are 65 and 117 where 117 > 65.
Applying Euclid's division lemma, 

117 = 65 × 1 +52   ........(1)

The remainder is not 0 so we apply the process again on the numbers 65 and 52.

65 = 52 × 1 + 13  .......(2)

The remainder is not 0 so we apply the process again on the numbers 65 and 52.

52 = 13 × 4 + 0

The last non-zero remainder obtained was 13 which is the HCF of 65 and 117.
From (2) we get

65 = 52 × 1 + 13

⇒ 13 = 65 - 52 × 1

⇒ 13 = 65 - (117 - 65 ×1)  ......(From (1))

⇒ 13 = 65 - 117 + 65 × 1

⇒ 13 = 65 × 2 + 117 × (-1)

⇒ 13 = 65 × 2 - 117

On comparing it with 65n - 117 we get the value of n as 2.