What is the remainder when $4^{10}+6^{10}$ is divided by $25$? Without using calculator, how to decide? Must go with last two digits of $4^{10}+6^{10}$, can tell the last digit is $2$. How to tell the tenth digit of the sum? 
Thanks!

The last digit of $4^x$ is $4,6,4,6,4,6\dots \implies 4^{10}$ has last digit $6$
The last digit of $6^x$ is always $6 \implies 6^{10}$ has last digit $6$
Therefore $4^{10} + 6^{10}$ has last digit $2$.
Any multiple of $25$ has last digit $0$ or $5\implies
25$ cannot be a divisor! 
