Units Digits of Big Powers

Try solving this question:

What is the units digit of 13³⁵?

A) 1

B) 3

C) 5

D) 7

E) 9

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

13¹ = 13 (units digit is 3)

13² = 169 (units digit is 9)

13³ = 2197 (units digit is 7)

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 13² is the same as the units digit of 3², the units digit of 13⁵ is the same as the units digit of 3⁵, and so on.

Continuing , we get:

13¹ has units digit 3

13² has units digit 9

13³ has units digit 7

13⁴ has units digit 1

13⁵ has units digit 3

13⁶ has units digit 9

13⁷ has units digit 7

13⁸ 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 13ⁿ is 1

That is,

13⁴ has units digit 1

13⁸ has units digit 1

13¹² has units digit 1

13¹⁶ has units digit 1

. . . etc.

At this point, we can find the units digit of 13³⁵

Since 32 is a multiple of 4, 13³² must have units digit 1. From here, we’ll just continue the pattern:

13³² has units digit 1

13³³ has units digit 3

13³⁴ has units digit 9

13³⁵ has units digit 7

The units digit of 13³⁵ is 7, which means D is the answer to the original question.

For additional practice try these two questions: 

1. Find the units digit of 57³⁰

2. Find the units digit of 34³³

Answers below . . .

Answers to above questions:

1. 9

2. 4