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What is the units digit of (2x)^8?

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

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New post 03 Feb 2019, 23:47
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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.

_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html
4) Base while finding % increase and % decrease : https://gmatclub.com/forum/percentage-increase-decrease-what-should-be-the-denominator-287528.html


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

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New post 08 Feb 2019, 08:10
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
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html
4) Base while finding % increase and % decrease : https://gmatclub.com/forum/percentage-increase-decrease-what-should-be-the-denominator-287528.html


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

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New post 04 Feb 2019, 00:09
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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
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What is the units digit of (2x)^8?  [#permalink]

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New post 04 Feb 2019, 00:26
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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Re: What is the units digit of (2x)^8?  [#permalink]

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New post 07 Feb 2019, 21:41
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

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Re: What is the units digit of (2x)^8?   [#permalink] 07 Feb 2019, 21:41
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