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 57^k 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