2^3=8, and 8 divided by 7 leaves a remainder 1. This means that (2^3)^2=2^6 will also have remainder 1, and 2^99 divided by 7 has a remainder of 1 so 2^100 has a remainder of 2*1=2 when divided by 7.
3^6=729 which is 104 rem 1 when divided by 7. So 3^96 also has rem 1 when divided by 7. 3^4=81 which is 11*7 + 4 so 3^100 divided by 7 has rem 4.
4^3=2^6 which has rem 1 when divided by 7; 4^96 also has rem 1 and 4^4=256 which has rem 4 as does 4^100.
5 is 7-2 so that's rem -2; 25 has rem (-2)^2=4; 125 has rem (-2)^3=-8 which has rem -1; (-2)^6 has rem 1. Check: 5^6=15625=7*2232 + 1. So 5^96 gives rem 1 and 5^4 gives rem (-2)^4=16 which is rem 2.
(SUMMARY: Each of the numbers 2, 3, 4 and 5 raised to the power of 6 has a remainder of 1 when divided by 7, so since 6 goes into 100 16 times (6*16=96) each of the numbers raised to the power of 96 also has a rem of 1 when divided by 7. Therefore, we only had to take the numbers 16, 81, 256 and 625 (4th powers of 2, 3, 4 and 5) to get to the 100th power, and then we look at the divisibility of 7 into each of these, arriving at 2, 4, 4 and 2 as the remainders, the sum of which gives us the remainder we're looking for when further dividing it by 7.)
Now we just add the rems: 2+4+4+2=12 which has a rem of 5 when divided by 7. The whole original expression 2^100+3^100+4^100+5^100 then has rem 5 when divided by 7.