2000 AMC 10 Problem 6

The Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, \ldots starts with two 1s, and each term afterwards is the sum of its two predecessors. Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci sequence?

\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 9

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.