Can the last digit in the base ten expansion of X^2X be 4, whenever X is a positive integer?

If so, give an example. If not, prove it.

I do not believe it is possible.

X must end in a 2, 4, or 8 for any power of X to end in a 4. 

If X ends in 2 or 8, X^n will end in a 4 if n is 2 mod 4.  However, if X is 2 or 8, then 2X will always be 0 mod 4, so these won't work.

Similarly, if X ends in 4, X^n will end in a 4 if n is odd.  However, if X is 4 then 2X will obviously be even, so this won't work either.

 

 

Posted by tomarken on 2014-03-14 10:01:09