Difficulty: Easy
Correct Answer: 360
Explanation:
Introduction:
This problem tests your understanding of how to find the least common multiple (LCM) of several numbers, which is a common topic in aptitude exams. You need to identify the smallest positive integer that is divisible by all the given numbers 24, 36, and 40.
Given Data / Assumptions:
Concept / Approach:
To find the LCM using prime factorisation, we express each number as a product of prime factors. Then we take every prime that appears in any of the numbers, with the highest power that appears among them. Multiplying these highest powers gives the LCM.
Step-by-Step Solution:
Prime factorise each number: 24 = 2^3 × 3 36 = 2^2 × 3^2 40 = 2^3 × 5 Collect all prime factors: 2, 3, and 5 For 2: highest power among the three numbers is 2^3 For 3: highest power is 3^2 For 5: highest power is 5^1 Therefore, LCM = 2^3 × 3^2 × 5 Compute step by step: 2^3 = 8 3^2 = 9 8 × 9 = 72 72 × 5 = 360 So the LCM is 360
Verification / Alternative check:
We can check divisibility: 360 ÷ 24 = 15, 360 ÷ 36 = 10, and 360 ÷ 40 = 9. All results are integers, so 360 is a common multiple. Also, there is no smaller positive integer that is divisible by 24, 36, and 40 together, because we already used the maximum prime powers.
Why Other Options Are Wrong:
120 and 240 are multiples of some of the numbers but not of all three. For example, 120 is not divisible by 36, and 240 is not divisible by 36 either. 480 is a common multiple but not the least, so it cannot be the LCM.
Common Pitfalls:
Students sometimes add the numbers or multiply them directly instead of using prime factors or factorisation. Another frequent error is to take the minimum powers of primes (which gives the HCF) instead of the maximum powers required for LCM. Carefully separating HCF and LCM methods avoids such confusion.
Final Answer:
The least common multiple of 24, 36, and 40 is 360.
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