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# INTEGER & MULTIPLE (DS) QUSTION

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Intern
Joined: 05 Jun 2014
Posts: 2
INTEGER & MULTIPLE (DS) QUSTION [#permalink]

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12 Feb 2017, 02:32
If n is an integer, what is the unit digit of 2^n?

a) n is a multiple of 6
b) n is a multiple of 4

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Intern
Joined: 02 Oct 2016
Posts: 35
Schools: HEC Dec '17
Re: INTEGER & MULTIPLE (DS) QUSTION [#permalink]

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12 Feb 2017, 02:48
1
KUDOS
Hi Farjin,

Unit digits of different powers of the integer 2 follow a specific cycle.
2^1 will have 2 as the units digit
2^2 will have 4
2^3 will have 8
2^4 will have 6
2^5 will have 2
2^6 will have 4
2^7 will have 8
2^8 will have 6.

If you look carefully, the cycle 2,4,8,6 is repetitive.
When you can break down the power to the form 4m, 4m+1, 4m+2, 4m+3, where m is any integer, you'll be able to find the units digit.
For example, 2^21 = 2^(20+1) = 2^ [4(5)+1]. In this case, you can ignore the part which is a multiple of 4 and focus on the +1 part.
Hence 2^1 will have the same units digit as 2^21.

Statement 1 says n is a multiple of 6.
Case 1: n = 6
2^6 will have 4 as the unit digit.

Case 2: n = 12
2^12 will have 6 as the unit digit.

You now have two values, so answer isn't unique.

Statement 2 says n is a multiple of 4
N can be 4, 8, 12, 16, 20....
Whatever the value of n, 2^n will have 6 as the units digit. Hence, the answer is unique and sufficient.
Intern
Joined: 05 Jun 2014
Posts: 2
Re: INTEGER & MULTIPLE (DS) QUSTION [#permalink]

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12 Feb 2017, 11:12
Thank you I understand this concept very clearly. Thank you.
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Joined: 16 Aug 2015
Posts: 5452
GMAT 1: 800 Q59 V59
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Re: INTEGER & MULTIPLE (DS) QUSTION [#permalink]

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14 Feb 2017, 16:30
Dear Farjin,

Regarding unit digits of repeated integer multiplication, it has a period as follow.

$$2^1 = 1$$ - Unit digit : 2
$$2^2 = 4$$ - Unit digit : 4
$$2^3 = 8$$ - Unit digit : 8
$$2^4 = 16$$ - Unit digit : 6
$$2^5 = 32$$ - Unit digit : 2
$$2^6 = 64$$ - Unit digit : 4
$$2^7 = 128$$ - Unit digit : 8
$$2^8 = 256$$ - Unit digit : 6

2,4,8,6 happen by repetition.

Therefore, B is the correction answer.
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Re: INTEGER & MULTIPLE (DS) QUSTION   [#permalink] 14 Feb 2017, 16:30
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