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GMAT Quant: Finding the Units Digits of Large Powers

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 = 3249 (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:

  1. What is the units digit of  83^75?
  2. 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

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