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

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answer is C.

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Detailed Solution

Given integers 408 and 1032 and HCF(408, 1032) is expressible in the form of

1032 m − 408 × 5.

Euclid's Division Lemma states that if two positive integers a and b exist, then there must be unique values of q and r that satisfy the formula a= bq + r, where 0 ≤ r < b.
Apply Euclid’s division lemma on 408, 1032

1032 = 408 × 2 + 216 … … (1)

∵

remainder

≠ 0,

apply Euclid’s division lemma on divisor 408 and remainder 216

408 = 216 × 1 + 192 … … . (2)

∵

remainder

≠ 0,

apply Euclid’s division lemma on divisor 216 and remainder 192

216 = 192 × 1 + 24............... (3)

∵

remainder

≠ 0,

apply Euclid’s division lemma on divisor 192 and remainder 24

192 = 24 × 8 + 0

∴

remainder

= 0

∴

HCF of 1032, 408 is 24.
Given that

H C F = 1032 m − 408 × 5

24 = 1032 m − 408 × 5 24 = 1032 m − 2040 1032 m = 24 + 2040 1032 m = 2064 m = 20641032 m = 2

Therefore, value of m is 2.
Hence the correct option is 3.

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

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