## The Units Digits of Big Powers

Try this one:

What is the units digit of ?

A) 1

B) 3

C) 5

D) 7

E) 9

Let’s begin by looking for a pattern as we increase the exponent.

= 13 (units digit is 3)

= 169 (units digit is 9)

= 2197 (units digit is 7)

Aside: As you can see, the powers increase quickly! So, it’s helpful to observe that we need only consider the units digit when evaluating large powers. For example, the units digit of is the same as the units digit of , the units digit of is the same as the units digit of , and so on.

Continuing along, we get:

has units digit 3

has units digit 9

has units digit 7

has units digit **1**

has units digit 3

has units digit 9

has units digit 7

has units digit **1**

Notice that a nice pattern emerges. We get: 3-9-7-**1**-3-9-7-**1**-3-9-7-**1**-…

As you can see, the pattern repeats itself every 4 powers. I like to say that the “**cycle**” equals **4**

Now that we know the cycle is 4, we can make a very important observation:

**Whenever n is a multiple of 4, the units digit of is **

**1**

That is,

has units digit **1**

has units digit **1**

has units digit **1**

has units digit 1

. . . etc.

At this point, we can find the units digit of

Since 32 is a multiple of 4, must have units digit **1**. From here, we’ll just continue the pattern:

has units digit **1**

has units digit 3

has units digit 9

has units digit 7

The units digit of is 7, which means D is the correct answer to the original question.

For additional practice try these two questions:

1. Find the units digit of .

2. Find the units digit of .

Answers below….

*Answers to the above questions:*

*1. 9*

*2. 4*

## 74 comments

senate on April 2nd, 2013 at 1:46 am

What is the unit digit of 2^2013

Brent Hanneson on April 2nd, 2013 at 7:19 am

2^1 has units digit 2

2^2 has units digit 4

2^3 has units digit 8

2^4 has units digit 6

2^5 has units digit 2

2^6 has units digit 4

2^7 has units digit 8

2^8 has units digit 6

.

.

.

So, the “cycle” equals 4

When the exponent is a multiple of 4, the units digit is 6.

For example 2^4 has units digit 6, 2^8 has units digit 6, etc,

So, 2^2012 has units digit 6, which means 2^2013 has units digit 2,

Cheers,

Brent

talkingdesktop.com on June 7th, 2013 at 8:14 am

What's up, just wanted to tell you, I loved this article. It was funny. Keep on posting!

Alba on July 10th, 2013 at 3:24 pm

It helped me a lot.

Thank you.

Nandy on July 22nd, 2013 at 2:55 am

Really useful.Thank a lot

Brent Hanneson on July 22nd, 2013 at 8:02 am

Thanks for taking the time to provide feedback!

Cheers,

Brent

rahulsethi on July 24th, 2013 at 2:07 am

Wats d answer of (4!)^96 plz help me m stucked

deepchnd on September 20th, 2013 at 6:20 am

6

shifana on July 25th, 2013 at 12:16 am

very goob explanation for unit digit

shifana on July 25th, 2013 at 12:19 am

ver good explanation for unit digit

sumit joshi on August 12th, 2013 at 8:04 am

how to calculate the factorial of any number in an easy methord......

Brent Hanneson on August 12th, 2013 at 8:26 am

I don't believe there exists an easy way to calculate factorials.

Having said that, you shouldn't have to calculate big factorials on the GMAT.

Cheers,

Brent

Sonal on August 24th, 2013 at 2:28 am

Sir, your example is good but i still feel there is very limited explanation provided for every step execution. Also, the pattern forming is difficult to find out as during an exam time. There is always a time limit and one can't spend a lot of time in multiplying values one after the other in order to divulge a patter.

Is there a better option to solve such questions. Please suggest.

Brent Hanneson on August 24th, 2013 at 7:45 am

I'm not aware of a faster approach than what I've described. Since this is a relatively common question type on the GMAT, you will need to become proficient at listing the units digits of various powers.

Cheers,

Brent

deepchnd on September 20th, 2013 at 6:16 am

very much thanq full to u

vishal on October 15th, 2013 at 10:11 am

plz help .....find d unit diigits of

(7^95 - 3^58).....how to solve d substraction? plz explain...

Brent Hanneson on October 15th, 2013 at 10:49 am

7^95 = ---3 and 3^58 = ---9

So, 7^95 - 3^58 = (---3) - (---9) = ---4

Cheers,

Brent

vishal on October 16th, 2013 at 1:15 am

how does dis 4 come?...?.?(---3 ) - (---9) =4???

Brent Hanneson on October 16th, 2013 at 1:04 pm

Test it out.

Take ANY number with units digit 3 and subtract ANY number with units digit 9. What's the units digit of the result?

Cheers,

Brent

jisha on October 17th, 2013 at 6:14 am

Thank you......really useful!!!!

shalini on October 20th, 2013 at 5:57 pm

2^214^302 when divided by 9 .Find the remainder??plz sir explain in detail .finding difficulty in understanding the concept of remainder theorem

Brent Hanneson on October 21st, 2013 at 11:16 am

Hi shalini,

This question requires a knowledge of modular arithmetic (which is not required for the GMAT).

So, unless I'm missing something very basic, this question is out of scope.

Cheers,

Brent

amit on November 1st, 2013 at 5:36 am

Please help me with solving questions like36472^123!

Arnab Dandapath on August 26th, 2015 at 7:04 am

Ans is 6

123! will definitely have 2 zeroes at the end which means it's divisible by 4.

2's cycle is 2,4,8,6

So ans is 6

Brent Hanneson on November 1st, 2013 at 10:36 am

Great question. I think you should post it to the Problem Solving forum.

Cheers,

Brent

AJ on November 10th, 2013 at 7:28 pm

For those mentioning time constraints and having to multiply this out, remember, you only really need to multiply the units digit, not the entire number to get the higher powers' units digit. Thanks for writing this, it helped me out so much!

susmitha on January 20th, 2014 at 1:56 am

Wat cud be the maximum value of q in the following equation?

5P9+3r7+2q8=1114

tanmoy on March 4th, 2015 at 1:07 am

the answer is 9

monika on January 21st, 2014 at 2:43 pm

this short trick helped me a lot.

I want to ask u 1 ques.-

87^87 divided by 88

mohit agrawal on August 9th, 2014 at 12:54 am

remainder is 88 as you can split it as (87^87+1-1)/87+1

mohit agrawal on August 9th, 2014 at 12:58 am

sorry remainder is 1

sharique on June 10th, 2015 at 2:34 pm

{(ax-1)^n }/a will always have remaineder +1 if remainder is even and a- 1 when n is odd .

For ur question, 87^87 /88 , think of it is (88-1)^87/88.thus the remainder is 88-1=87. The idea is to make both the nos same in numerator and denominator.

Also if u had 177^177 / 88, it could be written as{ (88*2 -1)^177}/88, thus eventually giving 88-1 ; 87 as reminder

sandeep patel on February 6th, 2014 at 12:47 pm

what is the last digit of139 to the power 99 .can any one answer me.its a CAT question.ans is 7 how is it possible

Brent Hanneson on February 6th, 2014 at 1:02 pm

Hi Sandeep,

Are you sure the answer is 7?

Let's examine a few powers of 139:

139^1 = 139

139^2 = ---1

139^3 = ---9

139^4 = ---1

139^5 = ---9

139^6 = ---1

As we can see, the units digit will equal either 1 or 9

Continuing the pattern, the units digit of 139^99 will be 9

Cheers,

Brent

akshay on February 17th, 2014 at 1:33 pm

how to solve for units place of (15 + root of 220)^82 ?

Brent Hanneson on February 17th, 2014 at 3:10 pm

That question is WAYYYYYY out of scope for the GMAT.

Cheers,

Brent

Rajat on February 23rd, 2014 at 1:24 pm

How about the units digit of factorials, say 25!

Please help.

Brent Hanneson on February 23rd, 2014 at 1:27 pm

For anything past 4!, the units digit will always be 0.

We know this because, when n > 4, n! will contain a 2 and a 5 in its product, which means n! is a multiple of 2 AND a multiple of 5, which means it's a multiple of 10 as well.

All multiples of 10 have 0 as their units digit.

Cheers,

Brent

Vamsee on April 1st, 2014 at 1:22 pm

hello sir...

what is the remainder when 5^2013 is divided by 13?

plzz..

jus explain it to me in detail...!

Brent Hanneson on June 22nd, 2014 at 7:02 pm

This question is beyond the scope of the GMAT

Cheers,

Brent

blass on August 9th, 2014 at 2:16 am

5 is remainder 5^5 * 5^8 ^251 =5^5 1^251 so 3125 /13 R is 5

mohnish on June 20th, 2014 at 12:46 pm

how to find unit digit of (78^55/12^11)+(84^78/16^15)

Brent Hanneson on June 22nd, 2014 at 7:04 pm

Where did you get this question? It requires far too many steps to be a legitimate GMAT question

Cheers,

Brent

pradeep sharma on June 21st, 2014 at 5:17 am

1.19^9.7+11.2^9.3+1.03^10.9

Brent Hanneson on June 22nd, 2014 at 7:05 pm

The GMAT would never require test-takers to evaluate this.

Cheers,

Brent

KOYEL SARKAR on July 14th, 2014 at 12:39 pm

hello sir..can you explain me what will be the unit digit in the product of 122^173

Harish Vasudev on July 17th, 2014 at 1:25 am

Isn't the unit digit of 57^30 = 7 and not 9 as you have given in your answer.

Please clarify

Harish Vasudev on July 17th, 2014 at 1:34 am

Shit, Just realized it made a mistake. It is indeed 9. Thanks for this helpful tip.

Mili on September 24th, 2014 at 12:22 pm

12^13^14^15^16 Unit digit of this??

jaygeswar darik on October 12th, 2014 at 1:08 pm

what is the last two digits of 17^1729

sheena on September 9th, 2015 at 1:29 pm

7

Aimee on November 9th, 2014 at 10:23 pm

So would 2^2014 be the same as the example of 2^2013???

Aimee on November 9th, 2014 at 10:32 pm

so would finding the units of 2^2014 the same concept as the 2^2013?

Brent Hanneson on November 10th, 2014 at 2:31 pm

Yes, the approach would be the same.

Cheers,

Brent

sekhar kar on December 17th, 2014 at 8:11 am

(5673)^42734 x (956)^2876 x (999)^7961 = what?

Brent Hanneson on January 21st, 2015 at 1:28 pm

This one is a little time-consuming to be an official GMAT question.

nayan survase on January 21st, 2015 at 2:12 am

find the units digit in the product 24647^117 and 45631^24647 is what

a)9 b)7 c)3 d)1

Brent Hanneson on January 21st, 2015 at 1:29 pm

If you post this question in the Math Problem-Solving forum, someone will provide a nice solution.

Cheers,

Brent

shankar on March 18th, 2015 at 8:31 am

very useful well explained

thank u

Alisha on May 9th, 2015 at 5:15 am

helpful but just wondering,what is 7^2014 . thanks

Brent on May 11th, 2015 at 10:34 am

7^1 = 7

7^2 = 49

7^3 = __3

7^4 = __1

7^5 = __7

7^6 = __9

7^7 = __3

7^8 = __1

So, the pattern repeats every 4 exponents.

Also, when the exponent is divisible by 4, the units digit is 1.

So, 7^4 = __1, 7^8 = __1, 7^12 = __1, etc.

So, 7^2012 = __1

Now continue the pattern

7^2013 = __7

7^2014 = __9

Cheers,

Brent

Manhar on May 26th, 2015 at 1:55 am

plz tell me what will be

(564)^64

Sayyeda Raza on June 12th, 2015 at 5:57 pm

So helpful thank you so much!

rishabh sharma on June 17th, 2015 at 11:27 am

Sir what is the unit digit of 2^2015

akhil on July 28th, 2015 at 2:12 am

this was extremely helpful for Me.....

ash on August 22nd, 2015 at 5:40 am

13^35

so i take it as 3^35

the powers of 3 follows 4 as sequence

3^1=3

3^2=9

3^3=27=> 7

3^4=81=>1

3^5=243=>3

as you could see the sequence repeats itself , hence the sequence for 3 is 4

now divide 35 by 4 , you'l get a reminder of 3

so for 3 in the sequence it is 7 i.e .; 3^3= 7 , hence the unit digit for 13^35 =7

SAURABH on September 15th, 2015 at 1:08 pm

what would be unit digit of 13^36 ?

ash on September 17th, 2015 at 9:56 am

in ur case i take the number to be

3^36 so the remainder would b a zero

for these cases take the largest power ... for 3 it will follow 4 cycle i.e.;

3^1=3

3^2=9

3^3=7

3^4=1

hence your unit digit for 13^36 is 1

money on September 10th, 2015 at 3:07 am

why we r using cyclicity of 4....why we not use 5....I don't understand. ..with use 5 it should be come same digit....bt it is not coming why?

Brent Hanneson on September 10th, 2015 at 11:32 am

The cycle is definitely 4 since the units digit repeats itself every 4 power increases.

Sahil Verma on September 15th, 2015 at 6:01 am

Find last digit of 3^5^7^9 +1

SAURABH on September 15th, 2015 at 1:06 pm

what if we divide by 4 and get remainder as 0.

jawad on October 6th, 2015 at 8:23 am

how to find the last digit of 7^2007

Brent Hanneson on August 27th, 2015 at 10:12 am

What conclusion are you referring to?

Cheers,

Brent