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What is the units digit of 367?

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by Max@Math Revolution » Sat May 21, 2016 4:40 pm
What is the units digit of 3^67?
A. 1
B. 3
C. 5
D. 7
E. 9

*An answer will be posted in 2 days.
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by Simon Nguyen » Mon May 23, 2016 2:30 am
The unit digits of 3-base powers follow this sequence: 3, 9, 7, 1
Indeed:
3^1 = 3 => unit digit is 3
3^2 = 9 => unit digit is 9
3^3 = 27 => unit digit is 7
3^4 = 81 => unit digit is 1
3^5 = 243 => unit digit is 3

If 3 is raised to 67th power, the unit digit numbers will repeat this sequence until it stops at 67th power. Because the remainder of the division of 67 by 4 is 3 (16*4 + 3 =67), the digit number of the 67th power will be the third number in the sequence.

So the answer is 7
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by Max@Math Revolution » Mon May 23, 2016 7:37 pm
We get (~3)^1=~3, (~3)^2=~9, (~3)^3=~7, (~3)^4=~1. Units digit repeats 39713. Then, 3^67=3^[4(16)+3] 3^3=~7. Hence, the units digit becomes 7, and the correct answer is D.
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by jain2016 » Tue May 24, 2016 10:06 am
Hi Experts,

Can you please explain the below part?

3 (16*4 + 3 =67)

Thanks,

SJ
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by DavidG@VeritasPrep » Tue May 24, 2016 11:31 am
jain2016 wrote:Hi Experts,

Can you please explain the below part?

3 (16*4 + 3 =67)

Thanks,

SJ
Think of it like this - the pattern for the unit's digit for base 3 goes as follows:

3^1 ---> 3
3^2 ---> 9
3^3 ---> 7
3^4 ---> 1

3^5 ---> 3
3^6 ---> 9
3^7 ---> 7
3^8 ---> 1

So the pattern repeats every four terms. Put another way, every time the exponent is a multiple of 4, the units digit is '1.' Now we can think of a multiple of 4 that's close to 67. (So think either 64, as the above example does, or 68. Let's use 68.)

3^65 ---> 3
3^66 ---> 9
3^67 ---> 7
3^68 ---> 1
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by jain2016 » Wed May 25, 2016 3:05 am
Hi David ,

Thanks you sir for your reply. All clear.

Thanks,

SJ
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by danielle07 » Sun Sep 03, 2017 1:47 pm
This explanation is very good and easy to follow that i also got D as the answer

The unit digits of 3-base powers follow this sequence: 3, 9, 7, 1
Indeed:
3^1 = 3 => unit digit is 3
3^2 = 9 => unit digit is 9
3^3 = 27 => unit digit is 7
3^4 = 81 => unit digit is 1
3^5 = 243 => unit digit is 3

If 3 is raised to 67th power, the unit digit numbers will repeat this sequence until it stops at 67th power. Because the remainder of the division of 67 by 4 is 3 (16*4 + 3 =67), the digit number of the 67th power will be the third number in the sequence.

So the answer is 7
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Re: What is the units digit of 367?

by Scott@TargetTestPrep » Sat Feb 08, 2020 12:45 pm
Max@Math Revolution wrote: ↑
Sat May 21, 2016 4:40 pm
 What is the units digit of 3^67?
A. 1
B. 3
C. 5
D. 7
E. 9

*An answer will be posted in 2 days.
The pattern of units digits of the base of 3 is 3-9-7-1. Thus, 3^68 has a units digit of 1, so 3^67 has a units digit of 7.

Answer: D

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