Power Cycles

• Look only at the last three digits of 2ⁱ, with i=1,2,3,...
  Eventually the series starts to repeat earlier values. At what value of i does it repeat, and how long is the cycle?
• Do the same for the last three digits of kⁱ, (2 < k < 20).
• Do the same for the last N digits of k^i, (1 < N < 5).

Source: Original. Based upon a puzzle from reader Jimmy Chng Gim Hong.