What is the remainder when 4^96 is divided by 6?

Question

A. 0
B. 1
C. 2
D. 3
E. 4

vaibhav1221 · Accepted Answer

Remainder when  4^1 = 4  is divided by  6 = 4  
Remainder when  4^2 = 16  is divided by  6 = 4  
Remainder when  4^3 = 64  is divided by  6 = 4  
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.
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Remainder when  4^{96}  is divided by  6 = 4

Archit3110 · Answer

Solve it using the cyclicity rule which for 4 is : 4,6 ....
so at 4^96 we would get units digits ending with 6.. upon checking for any even power raised to 4 i.e 4^ even power and divided by 6 we always get remainder as 4 eg 4^2, 4^4 .... so 4^96 will also give remainder as 4 , so option E

chanchal1311 · Answer

GMATinsight  sir can you shed some light on this

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s111 · Answer

GMATinsight

Cyclicity of 4 is 2 ( 4,16,64...)

Means units digit of 4^96 is 6

And on division by 6 it should leave a reminder zero.

But in this case the answer is 4.

Posted from my mobile device

Archit3110 · Answer

You are seeing the remainder of units digit not whole value.. which is why you are not getting 4 as answer

s111 · Answer

BUT we always see units digit only to solve for reminders? please correct me here if i am wrong

487^191 divided by 5
cyclicity of 7 is 7,9,3,1 = 4
dividing 191/4 we get 47 pairs so units digit is 3

so reminder is 3

Archit3110 · Answer

NO.. PLEASE CHECK WHAT WOULD BE REMAINDER WHEN 4^2 IS DIVIDED BY 6 .. WILL YOU GET 0 OR 4..

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GMATinsight · Answer

Archit3110

This is NOT a question of unit digit calculation. this instead is a question of cyclicity of remainders

4^1  divided by 6 leaves remainder = 4
 4^2  divided by 6 leaves remainder = 4
 4^3  divided by 6 leaves remainder = 4
 4^4  divided by 6 leaves remainder = 4

i.e. remainder is always 4 irrespective of any exponent of 4 when the number  4^x  where m ≥1 is divided by 6

Answer: Option E

I hope this helps!!!

GMATinsight · Answer

Do NOT divide unit digit of number by 6 to calculate the remainder. Divide the entire number by 6 and then see the remainder.

Unit digit is reponsible for remainder only when the divisior is either 5 or 10

Archit3110 · Answer

GMATinsight  , I had understood the question , as rightly pointed out by you this is not a cyclcity question, I actually used cyclcity to determine the unit value of 4^96..but nonetheless it isn't of much use as remainder would always be' 4 'when divided by 6

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MayankSingh · Answer

I did this way:-

(4^96)/6= (4X4^95)/6

Above can be reduced to this, 2X (4^95)/3 {Keep a note we have cancelled 2 from neumerator & Denomenator so final result to be mutliplied by this}

Now, 2X 4^95/3 remainder= 2X (Rem. 4/3)^95=2 X (1)^95=2

Final Remainder= 2X2=4

Mansoor50 · Answer

GMATinsight 
Hi...am wondering if the following is correct:

4^96 = 2^192 = (3 - 1)^192
Now:

(3-1)^192/6 --> the '3' raised to any power will give a remainder of 3 when divided by 6 AND '1' will give a remainder of 1 when divided by 6, but since the minus sign has an even power we get:
r=3+1=4

your comments will be appreciated

GMATinsight · Answer

Mansoor50 Your reasoning is 100% correct here...

Darshi04 · Answer

Hello Mansoor50 I solved it the same way you did, with 1 exception. I think you created an extra step by writing 4^96 as 2^192, and further writing 2^192 as (3-1)^192. Instead, we can write 4^96 as (3+1)^96. Then remainder for 3/6 = 3 Remainder for 1/6 = 1 Final remainder = 3+1 = 4 Just shared this solution because I think it saves a little bit more time to do it this way.

rsrighosh · Answer

Is this 700 level question?

Bunuel · Answer

You can check difficulty level of a question along with the stats on it in the first post. For this question Difficulty = 700 Level. The difficulty level of a question is calculated automatically based on the timer stats from the users which attempted the question.

MathRevolution · Answer

4^{96} = (2^2)^{96}= 2^{192}

2^{4*48} = 2^4 = 16

=> 16 divided by 6 will give the remainder 4

Answer E

BrushMyQuant · Answer

We need to find the remainder when  4^{96}  is divided by 6?

4^{96}  =  (2^2)^{96}  =  2^{2*96}  =  2^{192}

To solve this problem we need to find the cycle of remainder of power of 2 when divided by 6

Remainder of  2^1  (=2) by 6 = 2
Remainder of  2^2  (=4) by 6 = 4
Remainder of  2^3  (=8) by 6 = 2
Remainder of  2^4  (=16) by 6 = 4

=> Cycle is 2

=> Odd Power remainder is 2
=> Even power remainder is 4
=> Remainder of  2^{192}  is divided by 6 = 4

So,  Answer will be E 
Hope it Helps!

Watch following video to MASTER Remainders by 2, 3, 5, 9, 10 and Binomial Theorem

https://www.youtube.com/watch?v=h1HTEhmUb1w

Natansha · Answer

How can we solve this ques using binomial theorem?

What is the remainder when 4^96 is divided by 6?

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What is the remainder when 4^96 is divided by 6?

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