Difficulty: Medium
Correct Answer: 5616
Explanation:
Introduction:
This question tests your ability to find the least common multiple (LCM) of several integers. LCM problems are widely used in aptitude exams to examine how well you work with prime factorization and multiples.
Given Data / Assumptions:
Concept / Approach:
The LCM of a list of numbers is obtained by taking the highest power of each prime that appears in the prime factorization of any of the numbers. We then multiply these highest powers together:
LCM = product of all primes raised to their maximum exponent appearing in any number
Step-by-Step Solution:
Step 1: Prime factorization of each number.16 = 2^424 = 2^3 * 336 = 2^2 * 3^252 = 2^2 * 1354 = 2 * 3^3Step 2: Take highest powers of all primes involved.For prime 2: maximum exponent = 4 (from 16).For prime 3: maximum exponent = 3 (from 54).For prime 13: maximum exponent = 1 (from 52).Step 3: Form the LCM.LCM = 2^4 * 3^3 * 13Step 4: Compute the value.2^4 = 163^3 = 2716 * 27 = 432432 * 13 = 5616
Verification / Alternative check:
Check divisibility: 5616 ÷ 16, 24, 36, 52 and 54 should all be integers. For example, 5616 ÷ 16 = 351, 5616 ÷ 24 = 234, 5616 ÷ 36 = 156, 5616 ÷ 52 = 108, and 5616 ÷ 54 = 104, all integers. Hence 5616 is a common multiple, and because we have used maximum prime powers, it is the least such multiple.
Why Other Options Are Wrong:
5618 and 5216: These are not constructed from the proper prime powers and will not be divisible by all the given numbers.
432: Although it is multiple of 16, 24 and 36, it is not divisible by 52 and 54.
10368: This is a multiple of 5616 and thus a common multiple but not the least one, so it cannot be the LCM.
Common Pitfalls:
Errors often occur when students miss one of the numbers or take the wrong highest power of a prime factor. Writing the factorization clearly and picking the maximum exponents systematically prevents such mistakes.
Final Answer:
The least common multiple of 16, 24, 36, 52 and 54 is 5616.
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