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Find the number of positive integers n, 1 <= n <= 100, such that n^3 + 5n^2 is the square of an integer.

 Jun 17, 2020
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This is \(n^2(n+5)\).

Because of that n^2 we know it's a square. We just need \(n+5\) to be a perfect square.

We get that the smallest n is 4, and we can go all the way up to n = 95. The number of perfect squares from 4 + 5 = 9 to 95 + 5 = 100 is 8.

 

You are very welcome!

:P

 Jun 17, 2020

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