If $10^{20} +20^{10}$ is divided by 4 then what would be its remainder? If $$10^{20} +20^{10}$$ is divided with 4 then what would be its remainder?

Since
$$
\begin{eqnarray*}
10^{20}+20^{10} &=&\left( 10^{10}\right) ^{2}+2^{10}10^{10} \\
&=&10^{10}\left( 10^{10}+2^{10}\right)  \\
&=&2^{10}5^{10}\left( 2^{10}5^{10}+2^{10}\right)  \\
&=&2^{10}2^{10}5^{10}\left( 5^{10}+1\right)  \\
&=&4^{10}5^{10}\left( 5^{10}+1\right)  \\
&=&4\left( 4^{9}5^{10}\left( 5^{10}+1\right) \right) ,
\end{eqnarray*}
$$
the remainder would be is $0$, because
$$\frac{10^{20}+20^{10}}{4}=4^{9}5^{10}\left( 5^{10}+1\right).$$
