# 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

1. 9

2. 4