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Math Expert V
Joined: 02 Sep 2009
Posts: 58417
What is the remainder when 4^96 is divided by 6?  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 57% (01:12) correct 43% (01:23) wrong based on 233 sessions

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

A. 0

B. 1

C. 2

D. 3

E. 4

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Re: What is the remainder when 4^96 is divided by 6?  [#permalink]

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1
1
Bunuel wrote:
What is the remainder when 4^96 is divided by 6?

A. 0

B. 1

C. 2

D. 3

E. 4

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 $$6 = 4$$
.
.
.
Remainder when $$4^{96}$$ is divided by $$6 = 4$$
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What is the remainder when 4^96 is divided by 6?  [#permalink]

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

A. 0

B. 1

C. 2

D. 3

E. 4

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

Originally posted by Archit3110 on 26 Nov 2018, 17:59.
Last edited by Archit3110 on 26 Nov 2018, 21:27, edited 1 time in total.
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Re: What is the remainder when 4^96 is divided by 6?  [#permalink]

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GMATinsight sir can you shed some light on this

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Re: What is the remainder when 4^96 is divided by 6?  [#permalink]

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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.

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Re: What is the remainder when 4^96 is divided by 6?  [#permalink]

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s111 wrote:
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

You are seeing the remainder of units digit not whole value.. which is why you are not getting 4 as answer
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Re: What is the remainder when 4^96 is divided by 6?  [#permalink]

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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
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Re: What is the remainder when 4^96 is divided by 6?  [#permalink]

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s111 wrote:
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

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

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Re: What is the remainder when 4^96 is divided by 6?  [#permalink]

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

A. 0

B. 1

C. 2

D. 3

E. 4

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

I hope this helps!!!
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Re: What is the remainder when 4^96 is divided by 6?  [#permalink]

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Archit3110 wrote:
s111 wrote:
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

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

Posted from my mobile device

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
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What is the remainder when 4^96 is divided by 6?  [#permalink]

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GMATinsight wrote:
Bunuel wrote:
What is the remainder when 4^96 is divided by 6?

A. 0

B. 1

C. 2

D. 3

E. 4

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

I hope this helps!!!

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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Re: What is the remainder when 4^96 is divided by 6?  [#permalink]

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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
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Re: What is the remainder when 4^96 is divided by 6?  [#permalink]

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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

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Re: What is the remainder when 4^96 is divided by 6?  [#permalink]

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Mansoor50 wrote:
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

Mansoor50

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Re: What is the remainder when 4^96 is divided by 6?  [#permalink]

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Mansoor50 wrote:
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

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. _________________
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We grow infinitely when we share. Re: What is the remainder when 4^96 is divided by 6?   [#permalink] 04 Jan 2019, 01:34
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