Difficulty: Medium
Correct Answer: 60
Explanation:
Introduction:
This problem combines the concepts of ratios, least common multiple (LCM), and highest common factor (HCF). You are given that three numbers are in a simple ratio and their LCM is known. From this, you must determine the HCF of the numbers.
Given Data / Assumptions:
Concept / Approach:
If three numbers are 3k, 4k, and 5k, where k is their common factor, then their HCF is k (because 3, 4, and 5 themselves are pairwise co prime). Their LCM will then equal 3 × 4 × 5 × k, because the LCM of 3, 4, and 5 is 60. Given the LCM value, we can solve for k and thus find the HCF.
Step-by-Step Solution:
Let the numbers be 3k, 4k, and 5k. Since 3, 4, and 5 are co prime, their LCM is 3 × 4 × 5 = 60. LCM of the actual numbers 3k, 4k, and 5k is therefore 60 × k. We are given that this LCM is 3600. So we have: 60 × k = 3600 Solve for k: k = 3600 / 60 = 60 Hence, the HCF of the three numbers is k = 60
Verification / Alternative check:
With k = 60, the numbers are 3 × 60 = 180, 4 × 60 = 240, and 5 × 60 = 300. The HCF of these numbers is clearly 60. The LCM of 180, 240, and 300 is 3600, which matches the given value. This confirms that the reasoning and computation are correct.
Why Other Options Are Wrong:
40, 100, and 120 do not satisfy the equation 60 × k = 3600 if treated as k. If they were the HCF, we would obtain inconsistent numbers or an incorrect LCM. Only k = 60 makes the ratio and the given LCM work together correctly.
Common Pitfalls:
Some learners mistakenly treat 3, 4, and 5 as the actual numbers rather than scaled versions. Others confuse the relationship between LCM and HCF for ratio based problems. Remember that when numbers are expressed as multiples of k, the common factor k is typically the HCF, and the LCM scales accordingly.
Final Answer:
The highest common factor of the three numbers is 60.
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