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What is the remainder when 2^n is divided by 10?

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What is the remainder when 2^n is divided by 10?  [#permalink]

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New post Updated on: 15 Aug 2018, 09:52
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[Math Revolution GMAT math practice question]

What is the remainder when \(2^n\) is divided by \(10\)?

1) \(n\) is a positive multiple of \(2\)
2) \(n\) is a positive multiple of \(4\)

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Originally posted by MathRevolution on 15 Aug 2018, 07:46.
Last edited by chetan2u on 15 Aug 2018, 09:52, edited 1 time in total.
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Re: What is the remainder when 2^n is divided by 10?  [#permalink]

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New post 15 Aug 2018, 07:53
MathRevolution wrote:
[Math Revolution GMAT math practice question]

What is the remainder when \(2^\)n is divided by \(10\)?

1) \(n\) is a positive multiple of \(2\)
2) \(n\) is a positive multiple of \(4\)


I) n=2,4,6,8...
2n/10 remainder could be 4,8....
Insufficient
II) n=4,8,12..
2n/10 remainder could be 8,6....
Insufficient
Combining both, remainder could be again 8,6...
Answer E.
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Re: What is the remainder when 2^n is divided by 10?  [#permalink]

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New post 15 Aug 2018, 08:09
Is it 2 power n or 2 times n. Looks like it's 2^n

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Re: What is the remainder when 2^n is divided by 10?  [#permalink]

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New post 15 Aug 2018, 09:26
MathRevolution wrote:
[Math Revolution GMAT math practice question]

What is the remainder when \(2^\)n is divided by \(10\)?

1) \(n\) is a positive multiple of \(2\)
2) \(n\) is a positive multiple of \(4\)


I think the question should be \(2^n\).
Then from statement 1 n can take values 2,4,6,8... Then \(2^n\) will not give a unique remainder. Hence 1 is insufficient.
Form statement 2 n can take values 4,8,12,16... Then \(2^n\) will be \(2^4\), \(2^8\), \(2^{12}\)...(Cyclicity of 2 is 4) In all such cases, the remainder will be 6. Hence, 2 is sufficient.
B is the answer.
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Re: What is the remainder when 2^n is divided by 10?  [#permalink]

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New post 17 Aug 2018, 02:27
=>
Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

The remainder when \(2^n\) is divided by \(10\) is the units digit of \(2^n\).

Now, \(2^1 = 2, 2^2 = 4, 2^3 = 8, 2^4 = 16\), and \(2^5 = 32\).
So, the units digits of \(2^n\) have period \(4\):
They form the cycle \(2 -> 4 -> 8 -> 6\).
Thus, \(2^n\) has the units digit of \(6\) when \(n\) is a multiple of \(4\).
Condition 2) is sufficient.

Condition 1)
If \(n = 4\), then \(2^n = 2^4 = 16\) has the units digit of \(6\).
If \(n = 2\), then \(2^n = 2^2 = 4\) has the units digit of \(4\).
Since we don’t have a unique solution, condition 1) is not sufficient.

Therefore, B is the answer.

Answer: B
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Re: What is the remainder when 2^n is divided by 10?  [#permalink]

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New post 18 Aug 2018, 11:57
souvonik2k wrote:
MathRevolution wrote:
[Math Revolution GMAT math practice question]

What is the remainder when \(2^\)n is divided by \(10\)?

1) \(n\) is a positive multiple of \(2\)
2) \(n\) is a positive multiple of \(4\)


I) n=2,4,6,8...
2n/10 remainder could be 4,8....
Insufficient
II) n=4,8,12..
2n/10 remainder could be 8,6....
Insufficient
Combining both, remainder could be again 8,6...
Answer E.


The powers for 2 follow a cyclic behavior (not sure if that's the correct word) - so every 2^4n will end with a 6 in the end. Making statement 2 sufficient.
Re: What is the remainder when 2^n is divided by 10? &nbs [#permalink] 18 Aug 2018, 11:57
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