Prove 11^10 - 1 is divisible by 100

Assume 11^10 - 1 = 0 (mod 100) (i.e. 11^10 - 1 is divisible by 100). 11^10 - 1 = 0 (mod 100) 11^10 = 1 (mod 100) (11^2)^5 = 1 (mod 100) (121)^5 = 1 (mod 100) (21)^5 = 1 (mod 100) [the remainder of 121 divided by 100 is 21] (21^2)^2(21) = 1 (mod 100) (41)^2(21) = 1 (mod 100) 81(21) = 1 (mod 100) 1701 = 1 (mod 100) ∴ 11^10 - 1 is divisible by 100.