Difficulty: Easy
Correct Answer: 504
Explanation:
Introduction / Context:
The question asks for the least common multiple (LCM) of 72 and 84. LCM problems are very common in aptitude tests and are used in questions involving common cycles, meeting times, and common denominators in fractions. Prime factorization provides a clear and reliable way to compute the LCM.
Given Data / Assumptions:
Concept / Approach:
To find the LCM, we express each number as a product of prime factors. For each prime that appears in either factorization, we take the highest power that appears in any of the numbers. The LCM is the product of these highest powers. This guarantees that the resulting number is divisible by each original number and is the smallest such common multiple.
Step-by-Step Solution:
Prime factorization of 72: 72 = 2^3 * 3^2.
Prime factorization of 84: 84 = 2^2 * 3 * 7.
Take highest powers for each prime:
For 2: max(3, 2) = 3, so use 2^3.
For 3: max(2, 1) = 2, so use 3^2.
For 7: presence only in 84, so use 7^1.
LCM = 2^3 * 3^2 * 7 = 8 * 9 * 7.
Compute: 8 * 9 = 72, then 72 * 7 = 504.
Verification / Alternative check:
Check divisibility:
504 / 72 = 7, which is an integer.
504 / 84 = 6, which is an integer.
No smaller candidate in the options is divisible by both 72 and 84, confirming that 504 is indeed the least common multiple.
Why Other Options Are Wrong:
144, 252, 420, and 756 either fail to be divisible by both 72 and 84 or are not the smallest such multiple. For example, 252 is divisible by 84 but not by 72, and 756 is a larger multiple of 504 but not the least common multiple.
Common Pitfalls:
A common error is to multiply the numbers directly without reducing by common factors, producing a multiple but not the least common multiple. Another mistake is to forget the prime 7 from 84 or use too small a power of 2 or 3. Careful and complete prime factorization avoids these problems.
Final Answer:
504
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