Units Digit?

The way to solve this problem is obviously not multiplying 139 by itself 99 times, but to look for patterns. The units digit is the last digit or the ones digit in the answer. If we multiply the 9 in 139 by itself, we get 81. 1 is the units digit in that answer. In this case the units digit of 139*139 is 1. We can take 1 again and multiply by 9 again, as 9 is always the ones digit in 139, and now we get 9. The units digit in for 139^3 is 9. We can repeat these steps on and on, and you can see that the units digit will keep alternating between 9 and 1. The even powers have units digits of 1, while the odd powers have units digits of 9. As 139 is to the power of 99 (an odd number), the units digit is 9. This is the easiest way to do this problem, but there is another method for numbers with more than two units digits in a sequence or pattern.

$\begin{aligned} 139^{99} & \equiv (140-1)^{99} \equiv (-1)^{99} \equiv -1 \equiv \boxed{9} \pmod {10} \end{aligned}$