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INTEGER & MULTIPLE (DS) QUSTION

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INTEGER & MULTIPLE (DS) QUSTION [#permalink]

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New post 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

In here answer is [b]. But I have little confusion about this. would you like to explain me. please


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Re: INTEGER & MULTIPLE (DS) QUSTION [#permalink]

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New post 12 Feb 2017, 02:48
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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.

Coming to your question,
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.

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New post 12 Feb 2017, 11:12
Thank you I understand this concept very clearly. Thank you.

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Re: INTEGER & MULTIPLE (DS) QUSTION [#permalink]

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New post 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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