What is the units digit of (303^33) – (73^15)?

A. 4

B. 5

C. 6

D. 7

E. 8

[spoiler]**C.**[/spoiler]

**Solution:**

When integers are multiplied by themselves they follow a repeating pattern. 3, 3*3, 3*3*3, 3*3*3*3 then it repeats 3*3*3*3*3 – **3**, 9, 27, 81, 24**3**. The units digit pattern of 3 is 3, 9, 7, 1. In the first case we need to find the 33rd number in the pattern. Because it is a pattern of 4 you can divide the exponent by 4 and use the remainder to figure out where you are in the pattern. 33/4 has a remainder of 1 so we must be in the first place of the pattern: 3. 15/4 has a remainder of 3 so we must be in the 3rd spot of the pattern: 7

In order to find the units digit of the resulting difference you only need to pay attention to the units digits. However, if the first number is bigger but has a smaller units digit you have to “borrow” a 1. This is not 3-7. It is 13-7 or 6.