what is the remainder when 2^1000 is divided by 7??

list the powers of 2 and their remainders:

2^1 2
2^2 4
2^3 1
2^4 2
2^5 4
2^6 1
...
you can see a pattern. So see how many times the pattern is repeated in 1000 powers of 2, and how many powers are left over.