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

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

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Updated on: 15 Aug 2018, 09:52
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25% (medium)

Question Stats:

68% (01:07) correct 32% (01:06) wrong based on 59 sessions

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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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"Only $99 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" 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. Formatted question  Math Revolution Discount Codes Manhattan GMAT Discount Codes EMPOWERgmat Discount Codes Ask GMAT Experts Forum Moderator Status: Preparing for GMAT Joined: 25 Nov 2015 Posts: 927 Location: India GPA: 3.64 Re: What is the remainder when 2^n is divided by 10? [#permalink] ### Show Tags 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. _________________ Please give kudos, if you like my post When the going gets tough, the tough gets going... Intern Joined: 16 Jul 2018 Posts: 7 Re: What is the remainder when 2^n is divided by 10? [#permalink] ### Show Tags 15 Aug 2018, 08:09 Is it 2 power n or 2 times n. Looks like it's 2^n Posted from my mobile device Manager Joined: 18 Jul 2018 Posts: 192 Location: India Concentration: Finance, Marketing WE: Engineering (Energy and Utilities) Re: What is the remainder when 2^n is divided by 10? [#permalink] ### Show Tags 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. Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6227 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: What is the remainder when 2^n is divided by 10? [#permalink] ### Show Tags 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 _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$99 for 3 month Online Course"
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Re: What is the remainder when 2^n is divided by 10?  [#permalink]

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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...

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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