Aas8 * When $4^{m}+4^{n}$ is a multiple of 100, what is the minimum value of $m+n$? $(m, n$ are natural numbers, and $m>n$ )
Since $4^{m}+4^{n}=4^{n}\left(4^{m-n}+1\right), 100=4 \times 25$, so $n=1$.
The smallest natural number $k$ such that $4^{k}+1$ is divisible by 25 is 5. Therefore, $m \geqslant n+5 \geqslant 6 . m+n=7$ is the minimum value (when $n=1, m=6$, $4^{m}+4^{n}=4 \times 1025=4100$ is a multiple of 100).