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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 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 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.

**Practice Question:**

Q. What is the units digit of 83^{75}?

Ans: 6

Q. What is the units digit of 39^{61}?

Ans: 9

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