Mock AIME 2 2006-2007/Problem 3

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Problem

Let $\displaystyle S$ be the sum of all positive integers $\displaystyle n$ such that $\displaystyle n^2+12n-2007$ is a perfect square. Find the remainder when $\displaystyle S$ is divided by $\displaystyle 1000.$

Solution

If $n^2 + 12n - 2007 = m^2$ , we can complete the square on the left-hand side to get $n^2 + 12n + 36 = m^2 + 2043$ so $(n+6)^2 = m^2 + 2043$ . Subtracting $m^2$ and factoring the left-hand side, we get $(n + m + 6)(n - m + 6) = 2043$ . $2043 = 3^2 \cdot 227$ , which can be split into two factors in 3 ways, $2043 \cdot 1 = 3 \cdot 681 = 227 \cdot 9$ . This gives us three pairs of equations to solve for $n$ :