TITLE: How to find $3^{1000}\bmod 7$?
QUESTION [7 upvotes]: Sorry this is a really simple problem but I couldnt figure it out. I'm trying to find the value of $3^{1000}\bmod 7$. I know the actual value is 4, but I used a calculator. How would I simplify this problem to get the correct answer without a calculator? I dont want a direct answer, but maybe a nudge in the right direction, such as tricks I could use or maybe an alternate way of writing it.

REPLY [2 votes]: Since that 36mod7=1,
then we have 31000=36*166+4=34=4 mod7.