If the HCF of 408 and 1032 is expressible in the form 1032 m – 408 × 5, find m.

Given: HCF of 408 and 1032 is expressible in the form 1032 m – 408 × 5.

To find: The value of m.

Concept Used:

Euclid's division lemma:

If there are two positive integers a and b,

then there exist unique integers q and r such that,

a = bq + r where 0 ≤ r ≤ b.

Explanation:

As 1032 > 408,

By Euclid 's division algorithm,

1032 = 408×2 + 216

408 = 216×1 + 192

216 = 192×1 + 24

192 = 24×8 + 0

Since the remainder becomes 0 here, so HCF of 408 and 1032 is 24.

Now,

1032 m – 408× 5 = HCF of these numbers

⇒ 1032 m - 2040 = 24

⇒ 1032 m = 24+2040

⇒ m = 2064/1032

⇒ m = 2

So, value of m is 2.