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Problem 11 - Entrance Test

How many positive integers n satisfy that n²+3n is a perfect square?

Correct: C

n²+3n=k². Complete square: 4n²+12n=4k² → (2n+3)²−9=4k² → (2n+3)²−4k²=9 → (2n+3−2k)(2n+3+2k)=9. Factor pairs of 9: (1,9),(3,3). Solving gives n=1,3. So 2 values.