The Units Digits of Big Powers
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Brent is a GMAT Instructor with over 20 years of teaching experience. Click here to learn more about GMAT Prep Now's products and services. |
Try this one:
What is the units digit of
?
A) 1
B) 3
C) 5
D) 7
E) 9
Let’s begin by looking for a pattern as we increase the exponent.
= 13 (units digit is 3)
= 169 (units digit is 9)
= 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 is the same as the units digit of 
, the units digit of 
is the same as the units digit of 
, and so on.
Continuing along, we get:
has units digit 3
has units digit 9
has units digit 7
has units digit 1
has units digit 3
has units digit 9
has units digit 7
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 
is 1
That is,
has units digit 1
has units digit 1
has units digit 1
has units digit 1
. . . etc.
At this point, we can find the units digit of
Since 32 is a multiple of 4, 
must have units digit 1. From here, we’ll just continue the pattern:
has units digit 1
has units digit 3
has units digit 9
has units digit 7
The units digit of is 7, which means D is the correct answer to the original question.
For additional practice try these two questions:
1. Find the units digit of 
.
2. Find the units digit of 
.
Answers below….
Answers to the above questions:
1. 9
2. 4
2 comments
senate on April 2nd, 2013 at 1:46 am
What is the unit digit of 2^2013
Brent Hanneson on April 2nd, 2013 at 7:19 am
2^1 has units digit 2
2^2 has units digit 4
2^3 has units digit 8
2^4 has units digit 6
2^5 has units digit 2
2^6 has units digit 4
2^7 has units digit 8
2^8 has units digit 6
.
.
.
So, the “cycle” equals 4
When the exponent is a multiple of 4, the units digit is 6.
For example 2^4 has units digit 6, 2^8 has units digit 6, etc,
So, 2^2012 has units digit 6, which means 2^2013 has units digit 2,
Cheers,
Brent