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Math Expert V
Joined: 02 Aug 2009
Posts: 7949
What is the units digit of (2x)^8?  [#permalink]

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Difficulty:   65% (hard)

Question Stats: 45% (01:28) correct 55% (01:17) wrong based on 87 sessions

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What is the units digit of $$(2x)^8$$?
(1) The unit's digit of $$x^2$$ is 9.
(2) The unit's digit of $$x^3$$ is 7.

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Math Expert V
Joined: 02 Aug 2009
Posts: 7949
Re: What is the units digit of (2x)^8?  [#permalink]

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chetan2u wrote:
What is the units digit of $$(2x)^8$$?
(1) The unit's digit of $$x^2$$ is 9.
(2) The unit's digit of $$x^3$$ is 7.

When we talkalk of units digit, we should know the following rules of cyclicity..

every digit has cyclicity when it comes to last/unit's digit...each digit surely repeats the units digit after every 4th power..

i) 1, 5 and 6 and 0 repeat with each power..
1^1 or 1^2 will always give 1 and similarly for 5,6,0

ii) 4,9 repeat after increase of two powers..
4^1 or 4^3 or 4^5 will all leave 4 as last digit and 4^2 or 4^4 will leave 6 as last digits
so cyclicity is 4,6,4,6..
for 9 it is 9,1,9,1,9...

iii) remaining 2,3,7,8 repeat after every 4th power
so 2^1 , 2^(1+4), 2^(1+4+4) will leave 2 in each case ..cyclicity is 2,4,8,6,2,4,8,6..
for 3 it is 3,9,7,1,3,9,..
for 7 it is 7,9,3,1,7,9,3,1...
for 8 it is 8,4,2,6,8,4,2,6..

So now we can look at the question..
Units digit of (2x)^8...our answer will depend on x
x^8 will have same units digit as x^4...

(1) The unit's digit of $$x^2$$ is 9.
Now x has to be odd and we can see from above that 7^2 and 3^2 leave 9 as the units digit.
Easy to think that different possibilities of x and take it aa insufficient,but we are looking for x^4..
So here is another point to remember..
All odd digits have same units digit, that is 1, when raised to power 4
Thus both 3 and 7 raised to 4 will have 1 units digit
Sufficient

(2) The unit's digit of $$x^3$$ is 7
Again we can see that x can only be 3 and x^4 =7^4 that is the units digit is 1.
Sufficient.

D
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Re: What is the units digit of (2x)^8?  [#permalink]

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chetan2u wrote:
What is the units digit of $$(2x)^8$$?
(1) The unit's digit of $$x^2$$ is 9.
(2) The unit's digit of $$x^3$$ is 7.

From 1:- x can be 7 or 3 but unit digit of $$x^8$$ will always be one($$x^2$$ 4 times or check for both 7 and 3).
so we can find single solution.
From 3:- x can only be 7. so sufficient

Option D
General Discussion
Manager  G
Joined: 07 Aug 2017
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What is the units digit of (2x)^8?  [#permalink]

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1
chetan2u wrote:
What is the units digit of $$(2x)^8$$?
(1) The unit's digit of $$x^2$$ is 9.
(2) The unit's digit of $$x^3$$ is 7.

IMO D

$$(2x)^8 = 2^8*x^8$$

The unit digits of $$2^n$$ cycle as 2, 4, 6, 8, 0
The 8th power will have unit digit as 6.

Statement 1:
Unit digit of $$x^2$$ is 9.
This means the unit digits of $$x^n$$ cycle as 3, 9, 7, 1.
Therefore, the 8th power will have unit digit as 1. Multiplying 8th power of x to 8th power of 2 will give unit digit as 6.
Sufficient.

Statement 2:
Unit digit of $$x^3$$ is 7.
From this again we can derive the unit digits of $$x^n$$ cycle as 3, 9, 7, 1.
And again the unit digit of product of 8th power of x and 2 is 6.
Sufficient.

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Intern  Joined: 21 Mar 2018
Posts: 1
Re: What is the units digit of (2x)^8?  [#permalink]

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kunalcvrce wrote:
chetan2u wrote:
What is the units digit of $$(2x)^8$$?
(1) The unit's digit of $$x^2$$ is 9.
(2) The unit's digit of $$x^3$$ is 7.

From 1:- x can be 7 or 3 but unit digit of $$x^8$$ will always be one($$x^2$$ 4 times or check for both 7 and 3).
so we can find single solution.
From 3:- x can only be 7. so sufficient

Option D
Excellent

Sent from my Moto G (4) using GMAT Club Forum mobile app Re: What is the units digit of (2x)^8?   [#permalink] 07 Feb 2019, 22:41
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