What is the smallest positive integer n such that 2n is a perfect square and 5n is a perfect cube?

Guest Oct 4, 2021

#1**0 **

*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 = 20^{2} a perfect square

5n = 1000 = 10^{3} 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? 5^{3} 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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Guest Oct 5, 2021