What is the units digit of 248^20?

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What is the units digit of 248^20?

by Gmat_mission » Mon Jun 18, 2018 2:11 am

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What is the units digit of 248^20?

A. 0
B. 2
C. 4
D. 6
E. 8

[spoiler]OA=D[/spoiler].

How can I determine the correct answer in an easy and fast way? Could anyone give me some help? Please.

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by Vincen » Mon Jun 18, 2018 3:08 am
Hello Gmat_mission.

Let's take a look at your question.

First, let's rewrite the number 248 as follows $$248=2\cdot124=2\cdot2\cdot62=2\cdot2\cdot2\cdot31=2^3\cdot31.$$ Now, replacing this into the given expression we get $$\left(2^3\cdot31.\right)^{20}=2^{60}\cdot31^{20}.$$ Now, we can see that the units digit of the powers of 2 are:

2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 6
2^5 = 2
2^6 = 4
2^7 = 8
2^8 = 6
.
.
.

Hence, when the power is a multiple of 4 the units digit is 6. Since 60 is a multiple of 4, then the units digit of 2^60 is 6.

On the other hand, the units digit of the powers of 31 are:

31^1 = 1
31^2 = 1
31^3 = 1
.
.
.

It is always equal to 1.

Therefore, the units digit of (2^60)*(31^20) is equal to 6*1=6.

Hence, the correct answer is the option [spoiler]D=6[/spoiler].

I hope it can help you.

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by GMATGuruNY » Mon Jun 18, 2018 3:19 am
Gmat_mission wrote:What is the units digit of 248^20?

A. 0
B. 2
C. 4
D. 6
E. 8
When 8 is raised to consecutive powers, the unit digit repeat in a CYCLE OF 4:
8¹ --> units digit of 8.
8² --> units digit of 4. (Since the product of the preceding units digit and 8 = 8*8 = 64.)
8³ --> units digit of 2. (Since the product of the preceding units digit and 8 = 4*8 = 32.)
8� --> units digit of 6. (Since the product of the preceding units digit and 8 = 2*8 = 16.)
From here, the units digits will repeat in the same pattern: 8, 4, 2, 6.
Thus, the units digit repeat in a CYCLE OF 4.
Implication:
When an integer with a units digit of 8 is raised to a power that is a multiple of 4, the units digit will be 6.
Since 248²� has a units of 8 and an exponent that is a multiple of 4, the units digit = 6.

The correct answer is D.
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by Scott@TargetTestPrep » Wed Jun 20, 2018 4:05 pm
Gmat_mission wrote:What is the units digit of 248^20?

A. 0
B. 2
C. 4
D. 6
E. 8
We need to determine the units digit of 248^20, which is the same as the units digit of 8^20.

Let's determine the pattern of units digits for 8 raised to positive integer powers.

8^1 = 8 so units digit is 8

8^2 = 64 so units digit is 4

8^3 = 512 so units digit is 2

8^4 = 4096 so units digit is 6

Thus, we see that a base of 8 has a units digit pattern of 8-4-2-6, so when a base of 8 is raised to a power that is a multiple of 4, the units digit is 6.

Since 20 is a multiple of 4, 8^20, has a units digit of 6.

Answer: D

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