Difficulty: Medium
Correct Answer: 84
Explanation:
Introduction / Context:
This question involves HCF, LCM, and the ratio of two numbers. A key fact is that for two positive integers, the HCF cannot exceed the LCM. The statement in the question lists HCF and LCM in reverse order, but the only meaningful interpretation is that HCF = 21 and LCM = 84. Using the ratio 1 : 4, we can reconstruct the two numbers and then identify the larger one.
Given Data / Assumptions:
Concept / Approach:
If the ratio is 1 : 4, we can let the two numbers be x and 4x. The HCF of x and 4x is x, because x divides both and there is no larger common divisor than x itself. Using the given HCF, we set x = 21. Then the numbers become 21 and 84, and we can confirm the LCM from this pair.
Step-by-Step Solution:
Step 1: Let the two numbers be x and 4x based on the ratio 1 : 4.Step 2: HCF(x, 4x) = x.Step 3: The HCF is given as 21, so x = 21.Step 4: Therefore, the numbers are 21 and 84.Step 5: The larger number is 84.
Verification / Alternative check:
Check the LCM of 21 and 84: since 84 is a multiple of 21, LCM(21, 84) = 84, which matches the interpreted LCM. The ratio 21 : 84 simplifies to 1 : 4, also matching the given ratio. This confirms that the interpretation and resulting numbers are consistent.
Why Other Options Are Wrong:
48: If the larger number were 48 with ratio 1 : 4, the smaller number would be 12. HCF(12, 48) = 12, not 21.108: With ratio 1 : 4, the smaller number would be 27. HCF(27, 108) = 27, not 21.12: Too small to be the larger number in a ratio of 1 : 4 and does not match the HCF condition.96: This would imply the smaller number is 24, and HCF(24, 96) = 24, not 21.
Common Pitfalls:
Not recognizing that HCF cannot be greater than LCM and accepting the original order without correction.Failing to apply the ratio correctly when constructing the pair of numbers.Overlooking the fact that for numbers x and 4x, the gcd is always x.
Final Answer:
84
Discussion & Comments