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# Number Theory

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

Oct 4, 2021

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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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Oct 5, 2021