Three numbers are in the ratio 3 : 4 : 5 and their least common multiple (LCM) is 3600. Using the properties of ratios, LCM, and highest common factor (HCF), determine the HCF of these three numbers.

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:

• The three numbers are in the ratio 3 : 4 : 5.
• The LCM of the three numbers is 3600.
• All numbers are positive integers.

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