modulo in a 24 h clock Suppose we have a clock with $24$ hours and it's $12$ o'clock. I need to calculate what time it will be after $100^{100}$ hours.
Obviously I just have to calculate  $100^{100}=4 \pmod {24}$
So I started by $100=4 \pmod {24}$ but I cannot go anywhere from here.
I also tried typing it as $100=-20 \pmod {24}$ which didn't prove to be helpful either. Could someone help me with this one?

Many ways to approach this sort of problem. Mine may be a bit long, but it works.
You want $4^{100} = 2^{200} \pmod{24}$.
First note that $2^7 = 128 \equiv 8 = 2^3 \pmod{24}$
This allows you to write $2^{7k} \equiv 2^{3k} \pmod{24}$.
Now $200 = 7(28)+4$ so you can write $2^{200} \equiv 2^{3(28) + 4} \\= 2^{7(12)+4} \equiv 2^{3(12)+4} \\= 2^{40} \\= 2^{7(5)+5} \equiv 2^{3(5)+5} \\= 2^{20} \\= 2^{7(2)+ 6}\equiv 2^{12} \\= 2^{7+5} \equiv 2^{3+5} \\= 2^8 \\= 2^{7+1} \equiv 2^{3+1} \\= 16 \pmod{24}$
So the answer is $16$ hours after your start point.
