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Qus: If HCF (1008, 20) = HCF (20, a) = HCF (a, b) where 1008=20×q+a; 20=a×m+b where (q, a) and (m, b) are positive integers satisfying Euclid’s Division Lemma. What could be the values of a and b?

A. 24, 8
B. 20, 8
C. 10, 4
D. 8, 4

Solution

D. 8, 4

Explanation

If p=dÃ—q+r, (p>q) where p, q, d, r are integers and for a given (p, d), there exist a unique (q, r), then HCF (p, d) = HCF (d, r). Because this relation holds true, the Euclidâ€™s Division Algorithm exists in a step by step manner. So, to find the HCF (1008, 20), we use Euclidâ€™s division lemma at every step.

Step 1: 1008=20Ã—50+8 â‡’ HCF(1008, 20) = HCF(20,8) â‡’ a could be 8

Step 2: 20=8Ã—2+4 â‡’ HCF(20, 8) = HCF(8,4) â‡’ b could be 4

Step 3: 8=4Ã—2+0

HCF = 4

Since 1008=20Ã—q+a where q and a are positive integers satisfy Euclidâ€™s Division Lemma, we must have 0â‰¤a<20. So a is surely 8 and b is 4.