GMAT Quant: Finding the Units Digits of Large Powers

Raising to a power is iterated multiplication. Luckily, you can find your units digit with a simple multiplication pattern, even when you’re working with large powers. (For a refresh of the multiplication rules for unit digits, see our post on difficult units digits.)

See how you do with this question:

What is the units digit of 5745?

A) 1

B) 3

C) 5

D) 7

E) 9

 

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

571 = 57 (the units digit is 7)

572 = 3,249 (the units digit is 9)

573 = 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 572 is the same as the units digit of 72. Similarly, the units digit of 573 is the same as the units digit of 73.

So, once we know that the units of 572 is 9, we can find the units digit of 573 by multiplying 9 by 7 to get 63. So the units digit of 573 is 3.

To find the units digit of 574, we’ll multiply 3 by 7 to get 21. So the units digit of 574 is 1.

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

  • The units digit of 571 is 7
  • The units digit of 572 is 9
  • The units digit of 573 is 3
  • The units digit of 574 is 1
  • The units digit of 575 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 571 is 7

The units digit of 572 is 9

The units digit of 573 is 3

The units digit of 574 is 1

The units digit of 575 is 7

The units digit of 576 is 9

The units digit of 577 is 3

The units digit of 578 is 1

The units digit of 579 is 7

The units digit of 5710 is 9

The units digit of 5711 is 3

The units digit of 5712 is 1. . . etc.

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

Now to find the units digit of 5745, all we need to do is recognize that the units digit of 5744 is 1 (since 44 is a multiple of 4).

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

The units digit of 5744 is 1

The units digit of 5745 is 7

The units digit of 5746 is 9

The units digit of 5747 is 3 . . . etc.

So, the units digit of 5745 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 8375?
  2. What is the units digit of 3961?

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

 

 

 

 

Answers:

1. 7

2. 9

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  • Brent

    Brent Hanneson is a master tutor with over 20 years of teaching experience. He developed all the math content for Magoosh Test Prep. Brent plays ice hockey in his free time.