## If n and a are positive integers, what is the units digit of

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### If n and a are positive integers, what is the units digit of

by alanforde800Maximus » Wed Oct 26, 2016 6:36 am
If n and a are positive integers, what is the units digit of n^4a+2 - n^8a?
(1) n = 3
(2) a is odd.

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by [email protected] » Wed Oct 26, 2016 7:58 am
alanforde800Maximus wrote:If n and a are positive integers, what is the units digit of n^(4a+2) - n^(8a)?

(1) n = 3
(2) a is odd.

For statement 1, you'll want to establish the pattern for the units digit with base 3.

3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81

3^5 = units 3
3^6 = units 9
3^7 = units 7
3^8 = units 1

3^9 = units 3
3^10 = units 9
3^11 = units 7
3^12 = units 1

So notice the pattern. Anytime the exponent is a multiple of 4, the units digit will be 1. Increase the exponent by one, and the units digit will be 3, etc.

Let's look at the expression n^(4a+2) - n^(8a) one piece at a time.

First: n^(4a+2). Well, we know that n^(4a) will have a units digit of 1, as 4a would have to be a multiple of 4. (If a = 1, 4a = 4; if a = 2, 4a = 8, etc.) It follows that n^(4a + 1) would have a units digit of 3, and that n^(4a +2) will have a units digit of 9. So the first term has a units digit of 9.

Next: n^(8a). Well 8a is a multiple of 4, so we already know that the units digit for the second term will be 1.

(And we also can see that if the base is 3, the first term will be smaller than the second term. If a = 1, we have 3^6 - 3^8; If a = 2, we have 3^10 - 3^16, etc.) If we know that the first term ends in 9 and that the second term ends in 1, and we know that the second term is larger, we can find the units digit of the difference. (Think 9 - 11, or 19 - 31. The units digit is always the same.) This statement alone is sufficient.

Statement 2: Say a =1 and n = 1. n^(4a+2) - n^(8a) = 1^6 - 1^8 = 0, and 0 is our units digit.
Say a = 1 and n = 3. n^(4a+2) - n^(8a) = 3^6 - 3^8. We already know from our analysis in S1 that 3^6 has a units digit of 9 and 3^8 has a units digit of 1. 3^6 - 3^8 will have a units digit of 2. Because we get different results, this statement alone is not sufficient.

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by ceilidh.erickson » Mon Oct 31, 2016 8:27 am
Many GMAT questions that ask about units digits will require you to find the pattern for a particular integer base raised to the nth power.

Here are some more for you to practice with:
https://www.beatthegmat.com/what-is-the- ... tml#554073
https://www.beatthegmat.com/what-is-the- ... tml#544267
https://www.beatthegmat.com/if-n-and-m-a ... tml#544266
https://www.beatthegmat.com/if-r-s-and-t ... tml#548713
Ceilidh Erickson
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by [email protected] » Tue Nov 01, 2016 5:21 am
alanforde800Maximus wrote:If n and a are positive integers, what is the units digit of n^4a+2 - n^8a?
(1) n = 3
(2) a is odd.
We need to determine the units digit of n^(4a + 2) - n^(8a). We can start by simplifying n^(4a + 2) to (n^4a)(n^2). Thus, we need to find the units digit of:

(n^4a)(n^2) - n^(8a)

Statement One Alone:

n = 3

Since we know that n = 3, we can substitute 3 for n in our given expression.

So we need to determine the units digit of: (3^4a)(3^2) - 3^(8a)

Let's start by evaluating the pattern of the units digits of 3^n for positive integer values of n. That is, let's look at the pattern of the units digits of powers of 3.

3^1 = 3

3^2 = 9

3^3 = 27

3^4 = 81

3^5 = 243

The pattern of any base of 3 repeats every 4 exponents. The pattern is 3-9-7-1. In this pattern, all positive exponents that are multiples of 4 will produce a 1 as its units digit. Since the exponents 4a and 8a are multiples of 4, we know that 3^(4a) and 3^(8a) will produce a units digit of 1. We also can determine that 3^2 produces a units digit of 9, thus:

The units digit of (3^4a)(3^2) - 3^8a = 1 x 9 - 1 = 8.

Statement one alone is sufficient to answer the question. We can eliminate answer choices B, C, and E.

Statement Two Alone:

a is odd.

Only knowing that a is odd is not enough to determine the units digit of (n^4a)(n^2) - n^(8a). We need more information about the value of n to determine a definitive answer. Statement two alone is not sufficient. 