<s>[INST] 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!
 [/INST] $$4^4=256 \equiv 6 \mod (25)$$
$$ 4^8 \equiv 36 \equiv 11 \mod (25)$$
$$ 4^{10} \equiv 16\times 11= 176 \equiv 1 \mod (25)$$
$$6^2 =36 \equiv 11 \mod (25)$$
$$6^4 \equiv 121 \equiv -4 \mod (25)$$
$$6^8 \equiv 16 \mod (25)$$
$$6^{10} \equiv 176 \equiv 1 \mod (25)$$ 
$$4^{10} + 6^{10} \equiv 2 \mod (25)$$
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