If the H.C.F of 85 and 153 is expressible in the form of 85 m – 153, then find the value of m.

We would first calculate H.C.F of 85 and 153 by using Euclid's Division Lemma.

According to the Euclid's Division Lemma,

If we have two positive integers a and b, then there exist unique integers q and r which satisfies the condition a = b q + r where 0 ≤ r ≤ b. and q is the quotient and r is remainder.

Therefore,

153 = 85 × 1 + 68

85 = 68× 1 + 17

68 = 17× 4 + 0

Now the remainder is zero, thus the H.C.F of 85 and 153 is 17.

But it’s given that H.C.F of 85 and 153 can be expressed as 85m – 153

⇒ 17 = 85m – 153

⇒ 85m = 153 + 17

⇒ 85m = 170

⇒

⇒ m = 2