2019 AMC 10 A
Complete problem set with solutions and individual problem pages
What is the greatest threedigit positive integer for which the sum of the first positive integers is not a divisor of the product of the first positive integers? (2019 AMC 10A Problem, Question#9)
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The sum of positive integers is , and we want this not to be a divisor of (the product of the first positive integers). Notice that if and only if were composite, all of its factors would be less than or equal to , so would be able to cancel with these factors in , and thus the sum would be a divisor. Hence in this case, must instead be prime. The greatest threedigit integer that is prime is , so we subtract to get .
As in Solution , we deduce that must be prime. If we can't immediately recall what the greatest threedigit prime is, we can instead use this result to eliminate answer choices as possible values of .Choices , , and don't work because is even, and choice does not work since is divisible by . Thus, the correct answer must be .
