Problem:
What is the greatest three-digit positive integer for which the sum of the first positive integers is a divisor of the product of the first positive integers?
Answer Choices:
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Solution:
The sum of the first positive integers is , and the product of the first positive integers is . If is odd, then divides because is an integer between and . If is even, then does not divide if and only if is prime. Because is the greatest three-digit prime number, the greatest three-digit positive integer for which the sum of the first positive integers is not a divisor of the product of the first positive integers is .
The problems on this page are the property of the MAA's American Mathematics Competitions