If n and a are positive integers, what is the units digit of n^(4a+2)

VERITAS PREP OFFICIAL SOLUTION:

Question Type: What Is the Value? This question asks for the unit’s digit of n^(4a + 2) – n^(8a).

Given information in the question stem or diagram: n and a are positive integers.

Statement 1: In the Algebra lesson, you learned that all numbers have a repeating pattern of units digits when they are raised to certain powers. For instance, here is the progression for 3, which is given as the value for n in Statement 1:

Units Digit of 3^1 = 3
Units Digit of 3^2 = 9
Units Digit of 3^3 = 7
Units Digit of 3^4 = 1
Units Digit of 3^5 = 3
Units Digit of 3^6 = 9
Units Digit of 3^7 = 7
Units Digit of 3^8 = 1

As you can see, the pattern repeats every 4. This Data Sufficiency question is asking whether you can determine the units digit of n^(4a+2) – n^(8a). If you can determine the units digit of each of those terms, then you can calculate the difference between their units digits. In Statement 1, you learn that n is 3, and it seems like you must also know something about a to answer the question. However, a closer look at the exponents shows that it does not matter what value a is. Remember that 3 raised to any multiple of 4 will always end in 1 (for instance, 3^12 or 3^24), and 3 raised to any multiple of 4 + 2 (for instance 3^16 or 3^26) will always end in 9. Therefore, regardless of what a is, n^(4a+2) will end in 9 and n^(8a) will end in 1. Therefore you can answer the question, and the answer must be A or D. Note: This is only possible because the numbers before the exponent a are multiples of 4.

Statement 2: A quick look at this statement and it is clearly insufficient. You must know something about n to answer the question. Since most people do not think that Statement 1 is sufficient, this is an important “Why Are You Here?” statement. The good test-taker will at this point try a few odd and even values for a to see if it makes a difference. In doing that, you will quickly prove to yourself that you do not need to know anything about a in order for statement 1 to be sufficient. This is another classic example of “Why Are You Here?—Temptation,” and if you play the Data Sufficiency game properly—leveraging hints and deciding if information is really important—you can get this problem correct every time!