HMMT February

Algebra Difficulty 4.8 AIME Find the answer

Find all positive integers $n>1$ for which $\frac{n^{2}+7 n+136}{n-1}$ is the square of a positive integer.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Write $\frac{n^{2}+7 n+136}{n-1}=n+\frac{8 n+136}{n-1}=n+8+\frac{144}{n-1}=9+(n-1)+\frac{144}{(n-1)}$ . We seek to find $p$ and $q$ such that $p q=144$ and $p+q+9=k^{2}$ . The possibilities are seen to be $1+144+9=154,2+72+9=83,3+48+9=60,4+36+9=49,6+24+9=39$ , $8+18+9=35,9+16+9=34$ , and $12+12+9=33$ . Of these, $\{p, q\}=\{4,36\}$ is the only solution to both equations. Hence $n-1=4,36$ and $n=5,37$ .

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