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What is the units digit of 3^n?

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by M7MBA » Sat Apr 07, 2018 12:58 am
What is the units digit of 3^n?

(1) n = 2x + 1, where x is a positive integer.
(2) n = 2k - 1, where k is a positive integer.

The OA is the option E.

How can I show that the correct option is E? I don't know what should I do.
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Source: — Data Sufficiency |
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by Vincen » Sat Apr 07, 2018 2:04 am
M7MBA wrote:What is the units digit of 3^n?

(1) n = 2x + 1, where x is a positive integer.
(2) n = 2k - 1, where k is a positive integer.

The OA is the option E.

How can I show that the correct option is E? I don't know what should I do.
Hello M7MBA.

Let's take a look here.

(1) n = 2x + 1, where x is a positive integer.

- If x=1 then n=3, therefore $$3^n=3^3=27\ \Rightarrow\ \ units\ digit\ is\ 7.$$

- If x=2 then n=5, therefore $$3^n=3^5=243\ \Rightarrow\ \ units\ digit\ is\ 3.$$

Since we get two different values, this statement is NOT SUFFICIENT.

(2) n = 2k - 1, where k is a positive integer.

- If k=1 then n=1, therefore $$3^n=3^1=3\ \Rightarrow\ \ units\ digit\ is\ 3.$$

- If k=2 then n=3, therefore $$3^n=3^3=27\ \Rightarrow\ \ units\ digit\ is\ 7.$$

Since we get two different values, this statement is NOT SUFFICIENT.

Now, using both statements together we get: $$n=n\ \Rightarrow\ \ \ 2x+1=2k-1\ \Rightarrow\ \ k=x+1.$$

- If x=1 then k=2 and n=3, therefore $$3^n=3^3=27\ \Rightarrow\ \ units\ digit\ is\ 7.$$

- If x=2 then k=3 and n=5, therefore $$3^n=3^5=243\ \Rightarrow\ \ units\ digit\ is\ 3.$$

Since we get two different values, this case is NOT SUFFICIENT.

Therefore, the correct answer is the option E.

I hope it helps you.
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by Jay@ManhattanReview » Mon Apr 09, 2018 2:49 am
M7MBA wrote:What is the units digit of 3^n?

(1) n = 2x + 1, where x is a positive integer.
(2) n = 2k - 1, where k is a positive integer.

The OA is the option E.

How can I show that the correct option is E? I don't know what should I do.
We have to determine the units digit of 3^n.

Let's take each statement one by one.

(1) n = 2x + 1, where x is a positive integer.

=> n is an odd number. Let's see how. Say x = 1, then n = 3; say x = 2, then n = 5

Case 1: n = 3, then the units digit of 3^3 (= 27) = 7.
Case 2: n = 5, then the units digit of 3^5 (= 243) = 3.

No unique answer. Insufficient.

(2) n = 2k - 1, where k is a positive integer.


=> n is an odd number. Let's see how. Say k = 1, then n = 1; say k = 2, then n = 3

Case 1: n = 1, then the units digit of 3^1 (= 3) = 3.
Case 2: n = 3, then the units digit of 3^5 (= 27) = 7.

No unique answer. Insufficient.

(1) and (2) together:

Even combining the two statements will not help since n is still an odd number, rendering the unit digit of 3^n = 3 or 7.

No unique answer. Insufficient.

The correct answer: E

Hope this helps!

-Jay
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