The H.C.F. and L.C.M. of two numbers are 84 and 21 respectively. If the ratio of the two numbers is 1 : 4, then the larger of the…

Question

The H.C.F. and L.C.M. of two numbers are 84 and 21 respectively. If the ratio of the two numbers is 1 : 4, then the larger of the two numbers is

1. A. 12
2. B. 48
3. C. 84 (Correct answer)
4. D. 108

Correct Answer

Option C — 84

Detailed Solution & Explanation

The correct answer is 84.

Key Points

• With a ratio of 1 : 4, write the numbers as x and 4x, where x is their HCF.
• Taking the HCF as 21, the numbers are 21 and 84.
• Their LCM is then 84, and 21 × 84 = 1764 = HCF × LCM, so the pair is consistent.
• The larger of the two numbers is 84.

Note

• As printed the question gives HCF = 84 and LCM = 21, but an HCF can never be greater than the LCM of the same two numbers — the two values are transposed in the booklet. Read the correct way round, the working above gives the keyed answer of 84.

Additional Information

• The general relation worth memorising is HCF × LCM = product of the two numbers, which holds for any pair.
• A structural consequence explains why the printed values cannot be right as given: the HCF always divides the LCM, so it can never be the larger of the two.
• For numbers in a ratio a : b where a and b are co-prime, the numbers are aH and bH with H the HCF, and the LCM is abH. Here 1 : 4 gives numbers H and 4H with LCM 4H — which is why the LCM turns out to be the larger number itself.