What is the smallest positive integer n such that 2n is a perfect square and 5n is a perfect cube?
If n = 200
2n = 400 = 202 a perfect square
5n = 1000 = 103 a perfect cube
Now that we've found the obvious one, let's try to find a smaller one. Let's consider that perfect cube.
If n is smaller than 200, then 5n is smaller than 1000, therefore its cube root must be smaller than 10.
So that would leave only cubing 1 through 9 to try, to find one smaller than 10 by brute force. But wait....
The product of any number multiplied by 5 will end only with either a 0 or a 5.
Therefore, for the cube to end with a 0 or a 5, the cube root must end with a 0 or a 5.
We've found that 10 cubed works, and the only smaller number that ends with either a 0 or a 5 is 5.
So, will 5 cubed work? 53 is 125. That makes 5n = 125, therefore n = 25. So, is 2n a perfect square?
2n would equal 50. 50 is not a perfect square, so n = 25 fails. We conclude that 200 is the smallest n.
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