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 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} = 3,249 (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:

- What is the units digit of 83
^{75}? - 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*