offers hundreds of practice questions and video explanations. Go there now.
Sign up or log in to Magoosh GMAT Prep.

GMAT Quant: Finding the Units Digits of Large Powers

See how you do with this question:

What is the units digit of  57^45?

A) 1

B) 3

C) 5

D) 7

E) 9

 

To solve this, we’ll begin examining smaller powers and look for a pattern.

57^1 = 57 (the units digit is 7)

57^2 = 3249 (the units digit is 9)

57^3 = 185,193  (the units digit is 3)

 

Aside: Since these powers increase quickly, it’s useful to notice that we need only multiply the units digit each time. For example, the units digit of  57^2 is the same as the units digit of  7^2. Similarly, the units digit of  57^3 is the same as the units digit of  7^3.

 

So, once we know that the units of  57^2 is 9, we can find the units digit of  57^3 by multiplying 9 by 7 to get 63. So the units digit of  57^3 is 3.

To find the units digit of  57^4, we’ll multiply 3 by 7 to get 21. So the units digit of  57^4 is 1.

When we start listing the various powers, we can see a pattern emerge:

The units digit of  57^1 is 7

The units digit of  57^2 is 9

The units digit of  57^3 is 3

The units digit of  57^4 is 1

The units digit of  57^5 is 7

At this point, we should recognize that the pattern begins to repeat. The pattern goes: 7-9-3-1-7-9-3-1-7-9-3-1-…

Since the pattern repeats itself every 4 powers, we say that the “cycle” equals 4

Now comes an important observation:

The units digit of  57^1 is 7

The units digit of  57^2 is 9

The units digit of  57^3 is 3

The units digit of  57^4 is 1

The units digit of  57^5 is 7

The units digit of  57^6 is 9

The units digit of  57^7 is 3

The units digit of  57^8 is 1

The units digit of  57^9 is 7

The units digit of  57^10 is 9

The units digit of  57^11 is 3

The units digit of  57^12 is 1. . . etc.

As you can see, since the cycle = 4, the units digit of  57^k will be 1 whenever k is a multiple of 4.

Now to find the units digit of  57^45, all we need to do is recognize that the units digit of  57^44 is 1 (since 44 is a multiple of 4).

From here, we’ll just continue with our pattern:

The units digit of  57^44 is 1

The units digit of  57^45 is 7

The units digit of  57^46 is 9

The units digit of  57^47 is 3 . . . etc.

So, the units digit of  57^45 is 7, which means the answer is D.

 

If you’d like to practice, you can answer these two questions:

  1. What is the units digit of  83^75?
  2. What is the units digit of  39^61?

(The answers can be found at the very bottom of this post)

 

 

 

 

Answers:

1. 7

2. 9

By the way, sign up for our 1 Week Free Trial to try out Magoosh GMAT Prep!

32 Responses to GMAT Quant: Finding the Units Digits of Large Powers

  1. krishna June 25, 2016 at 1:32 am #

    can u tell me if we get a remainder 0 what we must do

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert June 25, 2016 at 5:19 am #

      Hi Krishna ๐Ÿ™‚

      A number with a units digit of zero raised to any power (except 0) will still have a units digit of 0. Let’s look at the smallest such number, 10, to see how that works.
      10^0 = 1
      10^1 = 10
      10^2 = 100
      10^3 = 1000

      As you can see, the units digit is always 0 except when for 10^0, which is equal to 1. This pattern is observed for any numbers with a units digit of 0.

      I hope this helps ๐Ÿ™‚

    • Amritpal June 26, 2016 at 5:14 am #

      If you get 0 in remainder you can put that number power as 4 and solve the problem.
      For Example: 8^64
      64/4 = Remainder 0.
      So you can put 8^4 that is 8*8*8*8

      64*64 eliminate the tenth digit that is 4*4 = 16
      So you will get its unit digit 6.

      Hope this witl help ๐Ÿ™‚

  2. zd May 30, 2016 at 5:24 am #

    thks helped alot

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert June 2, 2016 at 7:37 am #

      You’re welcome ๐Ÿ™‚ Happy studying!

  3. Swagata May 18, 2016 at 11:59 am #

    Hi Mike,

    I am little confused here.Practice question 1. 83^ 75
    I am getting an answer of 3 and not 7 because 3^5 = has unit digit 3 and it has a pattern of 3971..3971…3971…so 83^74 = will have unit digit 1 and 83^75 = will have unit digit 3
    Am I missing anything which is why i am getting the units digit as 3 and not 7?

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert May 18, 2016 at 10:36 pm #

      Hi Swagata,

      Happy to help! ๐Ÿ™‚

      So let’s visualize the pattern you already mentioned:

      3^1 = …3
      3^2 = …9
      3^3 = …7
      3^4 = …1

      83^74 will not have a units digit of 3 because 74 is not divisible by 4. 83^74 will actually have a units digit of 9 and that means that 83^75 will have a units digit of 7.

      I hope that clarifies! ๐Ÿ™‚

  4. Prajapati neel May 15, 2016 at 12:16 am #

    Well done.. Ste by step.. Calculaation..now Easy to solve any example..!!!! Thanks dude..!!

  5. Sam April 22, 2016 at 7:01 am #

    hi

    How to find unit digit of the sum 228^128 + 393^193 + 447^147 + 522^122.

    When we solve answer is coming 1.

    But in book it is given as 6. How it is possible?

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert April 26, 2016 at 2:46 am #

      Hi Sam ๐Ÿ™‚

      To find the units digit of this sum, we must first determine the units digit of the individual elements of the sum. We can figure out the units digit of the four elements (228^128, 393^193, 447^147, and 522^122) by following the method explained in this post. For each element, we can determine the cycle of units digits and
      then use that information to figure out the units digit for the given exponents. Let’s look at how to do this for the first element, 228^128.

      Since the units digit is 8, we can focus on the units digits of the first few powers of 8:

      8^1 –> 8
      8^2 –> 4
      8^3 –> 2
      8^4 –> 6
      8^5 –> 8

      We can see that the units digit has a cycle of 4. Since 128 is a multiple of 4, then we can say that the units digit of 228^128 is 6. Similarly, 3, 7, and 2 also follow cycles of 4, and we can find the units digits of the corresponding elements using the same method as described above. The resulting units digits are:

      393^193 –> 3
      447^147 –> 3
      522^122 –> 4

      Adding all of these units digits together, we get a sum of

      6+3+3+4 = 16

      As you can see, the sum of the units digits is 6 ๐Ÿ™‚

      Hope this helps!

  6. Charu April 5, 2016 at 9:17 am #

    Hi Mike,

    I am little confused here.
    practice question 1. 83^ 75
    The answer should be 3 and not 7.
    Am missing anything which is why i am getting the units digit as 3 and not 7?

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert April 18, 2016 at 10:44 am #

      Hi Charu,

      Let’s start with the units digits of the first few powers of 83:

      83^1 = 3
      83^2 = 9
      83^3 = 7
      83^4 = 1
      83^5 = 3

      So using this period of 4 that repeats, we can know that:

      83^72 has a units digit of 1
      83^73 has a units digit of 3
      83^74 has a units digit of 9
      83^75 has a units digit of 7

      I’m not sure what you are doing to get 3 instead, but hopefully this helps clarify! ๐Ÿ™‚

  7. sam February 3, 2016 at 6:29 am #

    thank you so much ………………..effective one

  8. Shanmei January 6, 2016 at 6:12 pm #

    ๐Ÿ™‚ Thank you for the step-by-step tutorial, Brent!

  9. sneha kumari September 27, 2015 at 5:52 am #

    Thank u so much for explaining the topic in a good way…so that now I can solve many more other questions

  10. Pk June 1, 2015 at 2:06 am #

    Nice way of explaining it. Thanks a lot!.. ๐Ÿ™‚

    • Rita Kreig
      Rita Kreig June 2, 2015 at 2:47 pm #

      You’re so welcome! ๐Ÿ™‚

      • Sarthak July 17, 2015 at 8:33 am #

        units place for 2787^(14^13)?

        • r0h1t August 5, 2015 at 7:16 am #

          cycle of 7 completes at 4 i.e units places repeats after that.
          7^1=7
          7^2=9
          7^3=3
          7^4=1

          14×13=182=180+2
          180 is multiple of 4 so unite’s place of (2787)^180=1
          now unite’s place of (2787)^2=9
          Ans=1×9=9

          • M December 30, 2015 at 4:44 am #

            It is 14^13 NOT 14*13!

  11. Sirisha January 13, 2015 at 4:08 am #

    I love this concept ๐Ÿ™‚

    • SSY February 6, 2015 at 6:31 pm #

      Yes, me too..

  12. santosh November 28, 2014 at 11:49 pm #

    very helpfull for solving such problems.. update some more shortcuts

  13. Akhil November 5, 2014 at 10:40 pm #

    Thanks it’s very helpful

    • Mike MแถœGarry
      Mike November 10, 2014 at 11:28 am #

      Dear Akhil,
      You are quite welcome! ๐Ÿ™‚ Best of luck to you!
      Mike ๐Ÿ™‚

  14. kohila October 18, 2012 at 7:31 am #

    your explanation is nice.. but i have a doubt that how to find the unit digit of 9^26. the possibilities of 9 is 9 and 1. so two possibities . 26/2 gives no reminder.. then how to find the unit digit..

    • Mike MแถœGarry
      Mike October 18, 2012 at 1:19 pm #

      Dear Kohila,
      I’m happy to help. ๐Ÿ™‚ Think about the pattern
      9^1 — unit’s digit of 9
      9^2 — unit’s digit of 1
      9^3 — unit’s digit of 9
      9^4 — unit’s digit of 1
      Thus, *odd* powers of 9 have a units digit of 9, and *even* powers of 9 have a units digit of 1. We know 26 is even, so 9^26 has a units digit of 1.
      Does this make sense?
      Mike ๐Ÿ™‚

  15. kamran April 19, 2012 at 9:25 am #

    an easier way to answer this is..
    once you find the cycle (in the original question the cycle is 4 from 7,9,3,1) then find the remainder of exponent divided by the cycle –> 45/4 –> remainder=1. the final answer can then be calculated by taking the units digit of the following calculation –> units digit in the original question^remainder. in this case, 7^1=7.units digit of 7 is 7. therefore answer is 7.

    using same procedure for 83^75, cycle for units digit 3 is 4 (from 3,9,7,1). then find the remainder of exponent divided by the cycle–>75/4–>remainder=3. take units digit of following calculation–>units digit in the original question^remainder=3^3=27. units digit of 27 is 7, therefore answer is 7.

    for 39^61, cycle for units digit 9 is 2 (from 9,1). then find the remainder of exponent divided by the cycle–> 61/2–>remainder=1. take units digit of following calculation
    –>units digit in the original question^remainder=9^1=9. units digit of 9 is 9. therefore answer is 9.

    • Chris Lele
      Chris April 19, 2012 at 1:46 pm #

      Hi Kamran,

      I think your way is very efficient. In this post, I think Brent was trying to explicate the principle by writing out all the powers of 57. You wouldn’t necessarily do this when solving the problem; you would follow an approach similar to yours.

      So thanks for sharing :).

    • weirdo2989 September 30, 2012 at 8:07 am #

      This is a great info. Thanks a tonne ๐Ÿ™‚

      • Brent Hanneson October 18, 2012 at 7:36 am #

        Thanks for the feedback!

        Cheers,
        Brent

    • Brent Hanneson October 18, 2012 at 7:36 am #

      Hi Kamran,

      Sorry for not responding earlier.
      I’ll echo what Chris said: the instruction was intended to help you find the pattern. As you suggest, we need not write out everything.

      Cheers,
      Brent


Magoosh blog comment policy: To create the best experience for our readers, we will only approve comments that are relevant to the article, general enough to be helpful to other students, concise, and well-written! ๐Ÿ˜„ Due to the high volume of comments across all of our blogs, we cannot promise that all comments will receive responses from our instructors.

We highly encourage students to help each other out and respond to other students' comments if you can!

If you are a Premium Magoosh student and would like more personalized service from our instructors, you can use the Help tab on the Magoosh dashboard. Thanks!

Leave a Reply