What is the units digit of 367?

This topic has expert replies
User avatar
Elite Legendary Member
Posts: 3991
Joined: 24 Jul 2015
Location: Las Vegas, USA
Thanked: 19 times
Followed by:36 members
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.

Newbie | Next Rank: 10 Posts
Posts: 2
Joined: 23 May 2016

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

User avatar
Elite Legendary Member
Posts: 3991
Joined: 24 Jul 2015
Location: Las Vegas, USA
Thanked: 19 times
Followed by:36 members

by [email protected] 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.

Master | Next Rank: 500 Posts
Posts: 313
Joined: 13 Oct 2015
Thanked: 2 times

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

User avatar
Legendary Member
Posts: 2663
Joined: 14 Jan 2015
Location: Boston, MA
Thanked: 1153 times
Followed by:128 members
GMAT Score:770

by [email protected] » 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
Veritas Prep | GMAT Instructor

Veritas Prep Reviews
Save $100 off any live Veritas Prep GMAT Course

Master | Next Rank: 500 Posts
Posts: 313
Joined: 13 Oct 2015
Thanked: 2 times

by jain2016 » Wed May 25, 2016 3:05 am
Hi David ,

Thanks you sir for your reply. All clear.

Thanks,

SJ

Senior | Next Rank: 100 Posts
Posts: 40
Joined: 30 Aug 2017

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

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 6365
Joined: 25 Apr 2015
Location: Los Angeles, CA
Thanked: 43 times
Followed by:26 members
[email protected] 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

Scott Woodbury-Stewart
Founder and CEO
[email protected]

Image

See why Target Test Prep is rated 5 out of 5 stars on BEAT the GMAT. Read our reviews

ImageImage