What is the units digit of 13^86?

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BTGmoderatorDC: Moderator; Posts: 7187; Joined: Thu Sep 07, 2017 4:43 pmFollowed by: 23 members

by BTGmoderatorDC » Sun Oct 22, 2017 2:11 pm

What is the units digit of 13^86?

A. 1
B. 3
C. 5
D. 7
E. 9

How will i get the correct Option in this problem? Can experts help me with this?

OA E

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Source: Beat The GMAT — Problem Solving | Sun Oct 22, 2017 2:11 pm

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Jay@ManhattanReview: GMAT Instructor; Posts: 3008; Joined: Mon Aug 22, 2016 6:19 amFollowed by: 34 members; Location: Grand Central / New York; Thanked: 470 times

by Jay@ManhattanReview » Mon Oct 23, 2017 4:03 am

lheiannie07 wrote:What is the units digit of 13^86?

A. 1
B. 3
C. 5
D. 7
E. 9

How will i get the correct Option in this problem? Can experts help me with this?

OA E

The units digit of 13^86 would be the same as that of the units digit of 3^86. It is the unit digit of the base that matters.

We know that

3^1 = 3; unit digit = 3
3^2 = 9; unit digit = 9
3^3 = 27; unit digit = 7
3^4 = 81; unit digit = 1

3^5 = 243; unit digit = 3 = Same as that of 3^1
3^6 = _ _ _9 ; unit digit = 9 = Same as that of 3^2
3^7 = _ _ _7 ; unit digit = 7 = Same as that of 3^3
3^8 = _ _ _1 ; unit digit = 1 = Same as that of 3^4

We see that after each interval of 4 of the exponent, the unit digit of 3^(exponent) follows the same cycle.

Thus, the unit digit of 3^86 = unit digit of 3^(4*21 + 2) = unit digit of 3^(cycles of 4 + 2) = unit digit of 3^2 = 9.

The correct answer: E

Hope this helps!

-Jay

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Mon Oct 23, 2017 5:01 am

If you're interested, I wrote an article on the same topic on finding the units digit of large powers: https://www.gmatprepnow.com/articles/uni ... big-powers

The article ends with two additional practice questions.

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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Jeff@TargetTestPrep: GMAT Instructor; Posts: 1462; Joined: Thu Apr 09, 2015 9:34 amFollowed by: 22 members; Location: New York, NY; Thanked: 39 times

by Jeff@TargetTestPrep » Thu Dec 14, 2017 5:57 pm

lheiannie07 wrote:What is the units digit of 13^86?

A. 1
B. 3
C. 5
D. 7
E. 9

Since we only care about units digits, we can rewrite the expression as:

3^86

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. When writing out the pattern, notice that we are ONLY concerned with the units digit of 3 raised to each power.

3^1 = 3

3^2 = 9

3^3 = 7

3^4 = 1

3^5 = 3

The pattern of the units digit of powers 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 the units digit. Thus:

3^84 has a units digit of 1, 3^85 has a units digit of 3, and 3^86 has a units digit of 9.

Answer: E

Jeffrey Miller
Head of GMAT Instruction
[email protected]

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