Problem:
The least common multiple of and is , and the least common multiple of and is . What is the least possible value of the least common multiple of and
Answer Choices:
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Solution:
If , then and , and the least common multiple of and is . If , then any prime factor of must also be a factor of both and , and thus the only possible value is . In this case, must be a multiple of and a divisor of , so or . Similarly, must be a multiple of and a divisor of , so or . It follows that the least common multiple of and must be a multiple of . When , and , the least common multiple of and is exactly .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions