What is the remainder when the number below is divided by $100$? What is the remainder when the below number is divided by $100$?
$$
1^{1} + 111^{111}+11111^{11111}+1111111^{1111111}+111111111^{111111111}\\+5^{1}+555^{111}+55555^{11111}+5555555^{1111111}+55555555^{111111111}
$$
How to approach this type of question? I tried to brute force using Python, but it took very long time.

HINT:
$$(1+10n)^{1+10n}=1+\binom{1+10n}1(10n)\pmod{100}\equiv1+10n$$
and $$(5+50n)^{1+10n}=5^{1+10n}+\binom{1+10n}1(50n)5^{10n}\pmod{100}$$
Now, $$5^{m+2}-5^2=5^2(5^m-1)\equiv0\pmod{100}\implies5^{m+2}\equiv25\pmod{100}$$ for integer $m\ge0$
$$\implies5^{1+10n}+\binom{1+10n}1(50n)5^{10n}\equiv25+(1+10n)(50n)25\pmod{100}$$
$$\equiv25+1250n$$ for $n\ge1$
For odd $n,$ $$(5+50n)^{1+10n}\equiv25+50\pmod{100}$$
