C = 2 + 2^2 + 2^3 +...+ 2^99 + 2^100
=( 2 + 2^2 + 2^3 + 2^4 + 2^5) + (2^6 + 2^7 + 2^8 + 2^9 + 2^10) +...+ (2^95 + 2^96 + 2^97 + 2^98 + 2^99 + 2^100)
=2.( 1 + 2^1 + 2^2 + 2^3 + 2^4) + 2^6.(1 + 2^1 + 2^2 + 2^3 + 2^4) +...+ 2^96.(1 + 2^1 + 2^2 + 2^3 + 2^4)
=2.31 + 2^6.31 +...+ 2^96.31
=31.(2+ 2^6 +...+ 2^96) chia hết cho 31
=>C chia hết cho 31.