2012/13 AHSMC Part 1 Problem 6

Problem 6 of 16

For how many integers $n \ge 2$ is the sum of the first $n$ positive integers a prime number?

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Solution

The sum of the first $n$ positive integers is $\dfrac{n(n+1)}{2}$ . Suppose $n$ is even. Then we must have either $\dfrac{n}{2} = 1$ or $n + 1 = 1$ . Both lead to $n = 2$ . Suppose $n$ is odd. Then we must have either $\dfrac{n+1}{2} = 1$ or $n = 1$ . However, $n = 1$ is not allowed by the hypothesis. The answer is (B).

University of Alberta

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