Prove that a square of a positive integer cannot end with $4$ same digits different from $0$ Prove that a square of a positive integer cannot end with $4$ same digits different from $0$.
I already proved that square of positive integer cannot end with none of digits $1,2,3,5,6,7,8,9$ using the remainder of division by $3,4,8,10$. Now problem is how to prove that this number cannot end with $4444$.

As $x^2=4444\equiv12 \pmod{ 16}$, so no squares can end in four $4$'s.
