Find the remainder of 2^100/12 : Problem Solving (PS)

hsinam3

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Find the remainder of 2^100/12 [#permalink]

Updated on: 31 May 2013, 15:29

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A

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C

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Difficulty:

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52% (01:38) correct

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Find the remainder of 2^100/12

A. 4
B. 2
C. 8
D. 1
E. none

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Originally posted by hsinam3 on 31 May 2013, 14:03.
Last edited by Bunuel on 31 May 2013, 15:29, edited 1 time in total.

RENAMED THE TOPIC.

L

Bunuel

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Re: Find the remainder of 2^100/12 [#permalink]

31 May 2013, 16:52

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Expert Reply

hsinam3 wrote:

Find the remainder of 2^100/12

A. 4
B. 2
C. 8
D. 1
E. none

Look for the pattern:
2^2=4 divided by 12 yields the remainder of 4;
2^3=8 divided by 12 yields the remainder of 8;
2^4=16 divided by 12 yields the remainder of 4;
2^5=32 divided by 12 yields the remainder of 8;
2^6=64 divided by 12 yields the remainder of 4;
2^7=128 divided by 12 yields the remainder of 8;
...

We have that the remainder is 8 when the power of 2 is even.

Answer: A.

Similar questions to practice:
what-is-the-remainder-when-3-243-is-divided-by-141050.html
what-is-the-remainder-of-the-division-2-56-by-126889.html
what-is-the-remainder-of-126493.html
what-is-the-remainder-when-18-22-10-is-divided-by-99724.html
what-is-the-remainder-when-32-32-32-is-divided-by-100316.html

Theory on remainders is HERE.
PS questions on remainders are HERE.
DS questions on remainders are HERE.

Hope it helps.
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BangOn

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Re: Find the remainder of 2^100/12 [#permalink]

01 Jun 2013, 01:18

3

Kudos

hsinam3 wrote:

Find the remainder of 2^100/12

A. 4
B. 2
C. 8
D. 1
E. none

(2^4)^25 = 16^25 = (12+4)^25
Using binomial to find remainder. (x +a)^n divided by x is a ^n
4^25 = (16)^12 * 4 = (12+4)^12 * 4
4^13 = (16)^6 * 4 = (!2+4)^6*4
4^7 = (16)^3 * 4 = (12+4)^3*4
4^4 = (16)*16 = 4*4 = 16

Hence remainder 4.

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Re: Find the remainder of 2^100/12 [#permalink]

28 Jul 2013, 12:17

1

Bookmarks

Simple one would be :

REM(2^100/12)

=> REM [ 16^25 /12]

Since REM(16/12) = 4

=> REM [ 4^25/12 ]
=> REM [ (4^5)^5/12 ]
=> REM [ (16*16*4)^5 /12]
=> REM[ 64^5/12]

Since REM[64/12] =4

=> REM (4^5/12) =4

lucky1829

Intern

Joined: 27 Jan 2015

Posts: 5

Re: Find the remainder of 2^100/12 [#permalink]

01 Apr 2015, 04:19

I have problems with this one.

2^100 / 12 = 2^100 / (2^2*3) = 2^98 / 3

I tried to find the remainder when 2^98 is divided by 3. I got the solution R = 1
Why can't I split the 12 up to 2^2 * 3?

Thanks

Lucky2783

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Posts: 425

Concentration: International Business, Technology

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Re: Find the remainder of 2^100/12 [#permalink]

01 Apr 2015, 04:26

1

Bookmarks

hsinam3 wrote:

Find the remainder of 2^100/12

A. 4
B. 2
C. 8
D. 1
E. none

\(2^{100} = 4^{50}\) and all +ive power of 4 always give remainder 4 when divided by 12.
so answer is A .

L

Bunuel

Math Expert

Joined: 02 Sep 2009

Posts: 92760

Re: Find the remainder of 2^100/12 [#permalink]

01 Apr 2015, 05:19

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Expert Reply

lucky1829 wrote:

I have problems with this one.

2^100 / 12 = 2^100 / (2^2*3) = 2^98 / 3

I tried to find the remainder when 2^98 is divided by 3. I got the solution R = 1
Why can't I split the 12 up to 2^2 * 3?

Thanks

Following post by Karishma should clarify this issue: when-2-is-divided-by-194911.html#p1504691
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Jsalim2710

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Joined: 01 Jun 2015

Posts: 4

Re: Find the remainder of 2^100/12 [#permalink]

13 Sep 2015, 00:07

2^100 can be rewritten as (12-10)^100. Expanding this expression using the binomial theorem, all terms will contain 12^100 except the last term, this means that all terms will be divisible by 12 except the last term. finding the remainder when (-10)^100 divided by 12.

10/12 remainder is 4.
100/12 remainder is again 4.

Therefore similarly (-10)^100 remainder will be 4.

ParmarKP

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Joined: 07 Jul 2014

Posts: 10

Re: Find the remainder of 2^100/12 [#permalink]

14 Feb 2016, 23:39

hsinam3 wrote:

Find the remainder of 2^100/12

A. 4
B. 2
C. 8
D. 1
E. none

2^100/12 = (2*2*2^98)/12 = 2^98/3

or, 2^98/3 = (3-1)^98 / 3

All the terms in the expression are divisible by 3 but the last which is 1^98 ( Check this piece on use of binomial theorem by Veritas prep https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2011/05 ... ek-in-you/ super easy and very handy for remainder Qs)

Hence it boils down to 1/3 and the remainder is 1.

But we are not done yet. Remember how we simplified the expression in the beginning by dividing the divisor by 4? In order to get the correct remainder we have to multiply what we got as answer by 4. (Check how 1/3 and 2/6 despite being the same fraction do not have the same remainder!)

Answer A

sauravpaul

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Joined: 01 May 2015

Posts: 32

Re: Find the remainder of 2^100/12 [#permalink]

14 Feb 2016, 23:54

Bunuel wrote:

Following post by Karishma should clarify this issue: when-2-is-divided-by-194911.html#p1504691

Oh that's where I got it wrong. My approach was:

2^100/12 = 2^98/3

Remainder when:
2^1/3: 2
2^2/3: 1
2^3/3: 2
2^4/3: 1
2^5/3: 2
2^6/3: 1
2^7/3: 2

So, it is clear that for ever even power of 2, the remainder is 1 when that even power of 2 is divided by 3.

Since 98 is also even power, remainder will be 1.

Now that I read Karishma's post, I understand that I now need to multiply this 1 by 4 (because I had initially divided by 4)! Tricky.

D

stonecold

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Re: Find the remainder of 2^100/12 [#permalink]

21 Mar 2016, 09:07

Alternatively from a binomial point of you=>
2^100 => (12+4)^25 => 12P+ 4^25
No consider 4^25 => 4^1 divided by 12 => remainder => 4
4^2 divided by 12 => remainder => 4
4^3 divided by 12 => remainder => 4
hence 4^25 => 12Q+4
hence the expression becomes => 12T+4 => 4 is the remainder
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law258

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Re: Find the remainder of 2^100/12 [#permalink]

17 Nov 2016, 11:43

Bunuel - I approached this question in the same manner you did. However, I don't understand why 2^1 is not included on your list.

Can you please elaborate as to why you went straight for 2^2 and so on...?

B

vitaliyGMAT

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Joined: 13 Oct 2016

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Re: Find the remainder of 2^100/12 [#permalink]

26 Nov 2016, 06:53

1

Kudos

hsinam3 wrote:

Find the remainder of 2^100/12

A. 4
B. 2
C. 8
D. 1
E. none

\(\frac{2^{100}}{12} = \frac{2^{100}}{2^2*3} = \frac{2^{98}}{3}\)

\(2= -1 (mod 3)\)

\(\frac{(-1)^{98}}{3} = 1 (mod 3)\)

Multiplying this by \(2^2\)

\(1*4=(mod 3*4) = 4 (mod 12)\)

Remainder \(4\).

B

Andrewcoleman

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Joined: 22 Jan 2020

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Re: Find the remainder of 2^100/12 [#permalink]

12 Jun 2020, 11:51

Bunuel wrote:

hsinam3 wrote:

Look for the pattern:
2^2=4 divided by 12 yields the remainder of 4;
2^3=8 divided by 12 yields the remainder of 8;
2^4=16 divided by 12 yields the remainder of 4;
2^5=32 divided by 12 yields the remainder of 8;
2^6=64 divided by 12 yields the remainder of 4;
2^7=128 divided by 12 yields the remainder of 8;
...

We have that the remainder is 8 when the power of 2 is even.

Answer: A.

Hope it helps.

i think you might have done a typo of 8 instead of 4.

Thanks for all the questions Bunuel!

keep up the good work!

L

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Re: Find the remainder of 2^100/12 [#permalink]

06 Jun 2021, 03:47

hsinam3 wrote:

Find the remainder of 2^100/12

A. 4
B. 2
C. 8
D. 1
E. none

value of remainder of 2^ even power (>=4) when divided by 12 ; 4
and remainder when divided by 2^ odd power (>=5) is 8

2^100 is even power so remainder would be 4
option A

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Re: Find the remainder of 2^100/12 [#permalink]

02 Jun 2023, 13:53

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Re: Find the remainder of 2^100/12 [#permalink]

02 Jun 2023, 13:53

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