Find HCF of 65 and 117 and find a pair of integral values of m and n. Such that HCF = 65m + 117n.

The largest possible number which divides the given numbers exactly without any remainder is called the HCF (Highest Common Factor).

Answer: HCF of 65 and 117 is 13, the values of m and n are: m = 2 and n = -1

HCF of 65 and 117 is the highest number that divides 65 and 117 exactly leaving the remainder 0.

Explanation:

Here we are using Euclid's Division Algorithm to find the HCF of 65 and 117.

117 = (65 × 1) + 52.

65 = (52 × 1) + 13

52 = (13 × 4) + 0

Therefore, the HCF of 65 and 117 is 13.

Now to calculate the values of m and n we need to work as:

13 = 65 + 52 × (-1)

13 = 65 + [117 + 65 × (-1)] × (-1)

13 = 65 + 65 × (-1) x (-1) + 117 × (-1)

13 = 65 × 2 + 117 × (-1)

Now compare this equation with "HCF = 65m + 117n",

Hence, m = 2 and n = -1

Therefore, the HCF of 65 and 117 is 13, the values of m and n are: m = 2 and n = -1.