The Units Digits of Big Powers

by on October 30th, 2012

Try this one:

What is the units digit of 13^35?

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

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

13^1 = 13 (units digit is 3)

13^2 = 169 (units digit is 9)

13^3 = 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 13^2 is the same as the units digit of 3^2, the units digit of 13^5 is the same as the units digit of 3^5, and so on.

Continuing along, we get:

13^1 has units digit 3

13^2 has units digit 9

13^3 has units digit 7

13^4 has units digit 1

13^5 has units digit 3

13^6 has units digit 9

13^7 has units digit 7

13^8 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 13^n is 1

That is,

13^4 has units digit 1

13^8 has units digit 1

13^12 has units digit 1

13^16 has units digit 1

. . . etc.

At this point, we can find the units digit of 13^35

Since 32 is a multiple of 4, 13^32 must have units digit 1. From here, we’ll just continue the pattern:

13^32 has units digit 1

13^33 has units digit 3

13^34 has units digit 9

13^35 has units digit 7

The units digit of 13^35 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 57^30.

2. Find the units digit of 34^33.

Answers below….

 

Answers to the above questions:

1. 9

2. 4

136 comments

  • What is the unit digit of 2^2013

    • 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

    • 2

    • This demonstration is really helpful. Appreciated!

    • Thanks I was just looking for this answer because in quiz bowl we had a question on 2^18 and what the ones place would be. I guessed right (4) but I wanted to know how you would know that without working it out, as we only get 20 seconds for each question 

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

  • It helped me a lot.
    Thank you.

  • Really useful.Thank a lot

    • Thanks for taking the time to provide feedback!

      Cheers,
      Brent
       

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

    • 6

    • 4

    • no its 6

    • 4

    • Can anyone help explain how we arrive at 6?

    • 4!= 4×3×2×1 = 24
      Thus, (4!)^96 = 24^96 = 6 as 96 is even so the power will be 6.

  • very goob explanation for unit digit

  • ver good explanation for unit digit

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

    • 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

  • 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.

    • 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

  • very much thanq full to u 

  • plz help .....find d unit diigits of
      (7^95 - 3^58).....how to solve d substraction? plz explain...

  • 7^95 = ---3 and 3^58 = ---9
    So, 7^95 - 3^58 = (---3) - (---9) = ---4
    Cheers,
    Brent

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

    • 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

  • Thank you......really useful!!!!

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

    • 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

  • Please help me with solving questions like36472^123!

    • 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

      :) :)

    • i didnt understand how it is 6?

    • 123 will not be divisible by 4

    • The digits in unit's place of (36472) different powers of 2 is 2,4,6,8,2,4,6,8,...

      period being 4

      remainder when 123 divided by 4 gives 3

      so the 3rd number in the sequence is 6

      so the digit in the unit's place of (36472)^123 is 6

    • The sequence is 2 4 8 6 as
      2^1= 2
      2^2=4
      2^3=8
      2^4=6
      2^5=2
      and so on....so ans is 8

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

    Cheers,
    Brent

  • 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!

  • Wat cud be the maximum value of q in the following equation?
    5P9+3r7+2q8=1114

    • the answer is 9

  • this short trick helped me a lot.
    I want to ask u 1 ques.-
    87^87 divided by 88

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

    • sorry remainder is 1

    • {(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

  • 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

    • 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
       

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

    • That question is WAYYYYYY out of scope for the GMAT.

      Cheers,
      Brent

    • root of 220 can be written as root of 2 *root of 11* root of 10=15(approx)
      (15+15)=30 .so, unit digit of 10*3 will be 0

      answer is 0

  • How about the units digit of factorials, say 25!
    Please help. 

    • 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

  • hello sir...
    what is the remainder when 5^2013 is divided by 13?
    plzz..
    jus explain it to me in detail...!

    • This question is beyond the scope of the GMAT

      Cheers,
      Brent

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

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

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

      Cheers,
      Brent

    • Answer would be 0

  • 1.19^9.7+11.2^9.3+1.03^10.9

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

      Cheers,
      Brent

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

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

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

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

    • The unit digit of this number is 0.
      This can also be written as 2×3×4×5×6.
      Bcz it contain a pair of 2 and 5 so the ans would be 0.

  • what is the last two digits of 17^1729

    • 7

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

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

    • Yes, the approach would be the same. 

      Cheers,
      Brent

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

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

    • = 3^(Rem(34/4))*6*9^(Rem(61/4))

      = 3^2*6*9^1=9*6*9=4*9=6

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

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

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

      Cheers,
      Brent

    • Ans is 7.

  • very useful well explained 
    thank u

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

    • 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   

  • plz tell me what will be 
    (564)^64

    • 4^Rem(64/4) = 4^0 = 1

  • So helpful thank you so much! ;)

  • Sir what is the unit digit of 2^2015

    • 2 to the power 2015 's unit digit is ?

      Ans :
      2 follows 4 digit pattern. so divide 2015 by 4 . You get 3 as the remainder. 
      When the remainder is 3 the unit digit of 2 will be 8.
      So the unit digit of 2^2015 is 8.

  • this was extremely helpful for Me.....

  • 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

    • what would be unit digit of 13^36 ?

    • 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 

  • 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?

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

  • Find last digit of 3^5^7^9 +1

    • 8

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

    • if u get 0 as remainder !! take the largest power cycle like here since ur dividin by 4 ur power cycle must be 4, so take the value you get for 4th power cycle . hope this helps thank you 

  • how to find the last digit of 7^2007

  • what is the units digit of (7^5)^7

  • Can you help me solve ( 3^61×6^51×7^63 )?

  • Please advise on this question 
    What is the unit digit of 8^2016
    Thanks

  • what is unit place digit in (572)^443

  • can we also find 10th digit????or only unit digit...i have a seperate method to find 10th digit also...but don't know how to check it is true or not?

    • I don't believe there's a nice way to check for the tens digit. Also, I've never seen an official GMAT question test this concept.

      Cheers,
      Brent

  • how to find the unit digit of 1!+2!+3!+...+99! ?

  • the unit digit of the sum of the factorial series

  • thank you sirr @Brent Hanneson your trick helped me a lot 
    cheers

  • Can u help me what is the answer to this
     What is the ones digit of 2^30

  • Why do some powers (2,3 etc) have 4 numbers in their cycle while other have 2 and 6,5 only have one???

  • Is taking out cycles important

  • Thanks for posting it . IT REALLY HELPED ME A LOT..........

  • WHAT IS THE UNIT DIGIT OF 4^2003?

  • can u give a explanation?

  • Hi Brent,  how do the rules get applied when you have to divide? IE:

    (706^20)/2  or (286^17)/3

    • Those questions would be beyond the scope of the GMAT

  • What is the point of unit digits? I understand how to figure out the answer to these patterns, but why? What do you use that for?

  • Find the units digit of 2^22*3^33*4^44*5^55

  • How to find units digit o (729)^739

  • Its answer?

  • The unit digit of 168^156

  • Can u answer the question that is what's the answer for 2 power 49.??? IS there any formula to find the value of a n umber to the power n????

  • What is 2013 to the 2013th power? Give me awnsure before 4 today.

  • What would be tha unit digit of 3^99–3^50 ...please explain

  • What is to predict the ones digit in the value of 4to the 6 power
    Plz help me!

  • Find the ones digit of 2^79 I know the answer is 6.044629098 I just want to know how to solve it the steps or work or how to get the answer?

  • Pls tell me how to find unit place of (4/7)^1024

  • How to find unit place of (4/7)^1024

  • It's easier to solve these using congruences. for instance for 7^99. Using Euler's theorem, 7^phi(10) = 1 (mod 10). phi(10) = phi(2)*phi(5) = 4. 

    99 / 4 = 24.xxx

    (7^4)^24 * 7^3 = 1*7*7*7 = 49*7 = 9*7 = 63 = 3 (mod 10).

  • What is the unit digit of 2010^2011

  • what is the tens place of 23^2017

  • what is th tens place of 23^2017

  • what is the tens place of 23^2017

  • What would be the last digit of 1458793^60?

  • Thanks a lot brent for this article
    I can't u understand this concept in my coaching class so I studied your article .
    please answer this question 888^222 + 222^888%5 
    please find the unit digit.

  • what is the unit digit of 74^998

  • how to solve the value of 3^99-3^50

  • Nice thanks for explanation

  • What conclusion are you referring to? 

    Cheers,
    Brent 

  • Its answer

  • What would be the units digit of 729^733

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