Difficulty: Medium
Correct Answer: 60
Explanation:
Introduction / Context:
This question connects ratios with LCM and HCF. When numbers are given in a simple ratio, we can treat the actual numbers as multiples of a common factor. Using this structure, along with the given LCM, we can determine the HCF. This is a standard pattern seen in many aptitude tests involving number properties.
Given Data / Assumptions:
Concept / Approach:
If the numbers are in the ratio 3 : 4 : 5, we can write them as 3k, 4k, and 5k, where k is a positive integer that represents the HCF. Since 3, 4, and 5 are pairwise co prime, the LCM of 3k, 4k, and 5k is 3 * 4 * 5 * k = 60k. The given LCM value allows us to solve for k, which is then the HCF of the three numbers.
Step-by-Step Solution:
Let the numbers be 3k, 4k, and 5k.
Since 3, 4, and 5 are co prime pairwise, LCM(3, 4, 5) = 3 * 4 * 5 = 60.
Therefore, LCM(3k, 4k, 5k) = 60k.
Given LCM = 3600, so 60k = 3600.
Solve for k: k = 3600 / 60 = 60.
Thus, HCF of the three numbers is k = 60.
Verification / Alternative check:
The actual numbers are 3 * 60 = 180, 4 * 60 = 240, and 5 * 60 = 300. The HCF of 180, 240, and 300 is clearly 60. Their LCM should be 60 multiplied by LCM of 3, 4, 5 which is 60 * 60 = 3600, matching the given LCM, so the reasoning is consistent.
Why Other Options Are Wrong:
40, 100, 120, and 20 do not satisfy the relationship LCM = 60 * HCF in this setup. Using any of these values as the HCF would produce an LCM that is different from 3600, contradicting the given condition.
Common Pitfalls:
A frequent mistake is to misunderstand the role of k and treat 3, 4, and 5 as the actual numbers rather than as a ratio pattern. Another error is to compute HCF and LCM independently without using the powerful structure that the ratio provides. Remember that for numbers in a co prime ratio, the LCM of the actual numbers is the product of the ratio terms multiplied by the HCF.
Final Answer:
60
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