Difficulty: Easy
Correct Answer: 60
Explanation:
Introduction / Context:
This question focuses on finding the least common multiple (LCM) of two commonly used integers, 15 and 12. LCM is an essential concept for working with fractions, common denominators, and problems involving synchronized events in arithmetic and algebra.
Given Data / Assumptions:
Concept / Approach:
The LCM of two numbers can be found via prime factorization. We factor each number into primes and then take all distinct primes with their highest exponent appearing in either factorization. Multiplying these together yields the LCM. Another method is to use the relationship LCM * HCF = product of the two numbers, but prime factorization is very clear here.
Step-by-Step Solution:
Prime factorization of 15: 15 = 3 * 5.
Prime factorization of 12: 12 = 2^2 * 3.
Collect highest powers of each prime: 2^2, 3, 5.
LCM = 2^2 * 3 * 5 = 4 * 3 * 5.
LCM = 12 * 5 = 60.
Verification / Alternative check:
Check that 60 is a multiple of both numbers:
60 / 15 = 4, which is an integer.
60 / 12 = 5, which is an integer.
There is no smaller positive number than 60 that is divisible by both, because any multiple of 15 that is also a multiple of 12 must include the full prime components 2^2, 3, and 5, giving at least 60.
Why Other Options Are Wrong:
12 and 30 are multiples of only one of the numbers or are too small to accommodate both factorizations fully.
45 and 90 are multiples of 15, but 45 is not divisible by 12, and 90 is larger than necessary, so it is not the least common multiple.
Common Pitfalls:
Common issues include simply multiplying the numbers without reducing by common factors, which often produces a common multiple but not the least one. Another mistake is forgetting to include the highest power of each prime when using prime factorization. Carefully listing prime factors avoids these errors.
Final Answer:
60
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