Prove that for any x in the reals 5 divides x if and only if 5 divides x squared

I think we can do this by taking the contrapositive of the statement.

if the statement is true then the contrapositive is also true:-

If, for any real x, 5 does not divide x^2 then 5 does not divide x.

Any number that 5 will divide into must end in 0 or 5 so the above is evidently true.

So the original statement is also true.