Number Theory

 

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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