Question

find the HCF of 65 & 117 & express it in the form of 65m+117n.

Answer 1

Among 65 and 117; 117 > 65

Since 117 > 65, we apply the division lemma to 117 and 65 to obtain

117 = 65 × 1 + 52 … Step 1

Since remainder 52 ≠ 0, we apply the division lemma to 65 and 52 to obtain

65 = 52 × 1 + 13 … Step 2

Since remainder 13 ≠ 0, we apply the division lemma to 52 and 13 to obtain

52 = 4 × 13 + 0 … Step 3

In this step the remainder is zero. Thus, the divisor i.e. 13 in this step is the H.C.F. of the given numbers

The H.C.F. of 65 and 117 is 13

From Step 2:

13 = 65 – 52 × 1 … Step 4

From Step 1:

52 = 117 – 65 × 1

Thus, from Step 4, it is obtained

13 = 65 – (117 – 65 × 1) × 1

⇒13 = 65 × 2 – 117

⇒13 = 65 × 2 + 117 × (–1)

In the above relationship the H.C.F. of 65 and 117 is of the form 65m + 117 n, where m = 2 and n = –1