Units Digits of Big Powers

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 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 above questions:

1. 9

2. 4