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