The [Fibonacci sequence](https://artofproblemsolving.com/wiki/index.php/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](https://artofproblemsolving.com/wiki/index.php/Digit) is the last to appear in the units position of a number in the Fibonacci sequence?
$\textbf{(A)} \ 0  \qquad \textbf{(B)} \ 4  \qquad \textbf{(C)} \ 6  \qquad \textbf{(D)} \ 7  \qquad \textbf{(E)} \ 9$
Note that any digits other than the units digit will not affect the answer.  So to make computation quicker, we can just look at the Fibonacci sequence in $\bmod{10}$:
$1,1,2,3,5,8,3,1,4,5,9,4,3,7,0,7,7,4,1,5,6,....$
The last digit to appear in the units position of a number in the Fibonacci sequence is $6 \Longrightarrow \boxed{\mathrm{C}}$.