Difficulty: Easy
Correct Answer: 360
Explanation:
Introduction / Context:
The question is about finding the least common multiple (LCM) of three integers: 24, 36, and 40. LCM problems test understanding of prime factorization, common multiples, and the idea of the smallest positive number that each given integer can divide without leaving a remainder.
Given Data / Assumptions:
Concept / Approach:
To find the LCM of several integers, we use prime factorization. Each number is written as a product of primes raised to powers. For LCM, we take each prime that appears in any of the factorizations and use the highest power of that prime. Multiplying these highest powers gives the required LCM. This ensures the result is divisible by all the original numbers and is the smallest such common multiple.
Step-by-Step Solution:
24 = 2^3 * 3
36 = 2^2 * 3^2
40 = 2^3 * 5
For prime 2, highest power is 2^3.
For prime 3, highest power is 3^2.
For prime 5, highest power is 5^1.
LCM = 2^3 * 3^2 * 5 = 8 * 9 * 5 = 360.
Verification / Alternative check:
Check divisibility of 360:
360 / 24 = 15, which is an integer.
360 / 36 = 10, which is an integer.
360 / 40 = 9, which is an integer.
Since 360 is divisible by all three numbers, it is a common multiple. Any smaller candidate would miss at least one required prime power, so 360 is the least such multiple.
Why Other Options Are Wrong:
120 and 240 are not divisible by 36 because they lack the factor 3^2.
480 and 720 are multiples of 360 but not the least common multiple, they are larger than necessary.
Common Pitfalls:
Students often mistakenly add or average the numbers, or they forget to use the highest power of each prime when computing the LCM. Another common error is to find any common multiple but fail to confirm that it is the smallest one. Working systematically with prime factorization prevents these mistakes.
Final Answer:
360
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