GMAT, GRE Units Digit With Exponents

What is the units digit of 615 - 74 - 93?
A) 8 
B) 7 
C) 6 
D) 5 
E) 4

Answer

1. The units digit of any integer raised to a power will follow a pattern. Let's see how powers of 6 play out.
   61 = 6 --> units digit of 6
   62 = 36 --> units digit of 6
   63 = 216 --> units digit of 6
   64 = 1,296 --> units digit of 6
   
2. Let's apply the same process to powers of 7.
   71 = 7 --> units digit of 7
   72 = 49 --> units digit of 9
   73 = 343 --> units digit of 3
   74 --> units digit of 1
   75 --> units digit of 7
   76 --> units digit of 9
   77 --> units digit of 3
   78 --> units digit of 1
   
3. The units digit of 9 raised to an integer exponent follows a definitive pattern as well.
   91 = 9 --> units digit of 9
   92 = 81 --> units digit of 1
   93 --> units digit of 9
   94 --> units digit of 1
   95 --> units digit of 9
   
4. Thus far you know that the units digit of 615 – 74 – 93 = units digit of 6 – units digit of 1 – units digit of 9.
5. Simplify the first two terms of this expression: units digit of 6 – units digit of 1 = units digit of 5
6. So here's where we're at now: units digit of 5 - units digit of 9.
7. Hard Part: Some will think that since 5-9=-4, so units digit of the entire expression will be 4. But this fails to consider that the left term could be larger than the right, resulting in a units digit of 6. For example:
   15-9=6 {left term is larger}
   155-99=56 {left term is larger}
   155-999=-844 {right term is larger}
   15-99=-84 {right term is larger}
8. The crucial question in determining whether the units digit of the final expression is a 6 or a 4 is whether the left expression is larger than the right. In other words, "is (615 - 74) greater than 93?". I like to think of a negative in a units digit equation as its distance from 10. In this case 10-4 = 6.
9. To be sure we're right, take an approximate guess at the value of each term. You know that 93 will be less than 1000, which is 103, so the question of whether the units digit is 6 or 4 really rests on whether 615 - 74 is greater than 1000 (in which case the left term will be larger than the right term and the units digit will be 6) or whether 615 - 74 is less than a thousand (in which case the right term will be larger than the left term and the units digit will be 4).
10. The test does not require long tedious calculations and these are not necessary here. It should be rather clear that 615 - 74 is greater than 1000, in which case the units digit will be 6, not 4.
11. Units digit of 5 - units digit of 9 = units digit of 6 since the left term (i.e., the one with a 5) is larger than the right term. The final answer is a units digit of 6.
12. Answer choice C .