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# What is the remainder when you divide 2^200 by 7?

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Intern
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What is the remainder when you divide 2^200 by 7?  [#permalink]

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Updated on: 17 Oct 2012, 03:22
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What is the remainder when you divide 2^200 by 7?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

my approach :

2^x has a cyclicity of 4
Therefore, Rem(200/4) = 0

Rem(2^0/7) =1

Am i missing something here?

OA is D

Originally posted by g3kr on 16 Oct 2012, 19:42.
Last edited by Bunuel on 17 Oct 2012, 03:22, edited 1 time in total.
Renamed the topic and edited the question.
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16 Oct 2012, 20:46
3
5
g3kr wrote:

What is the remainder when you divide 2^200 by 7?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

my approach :

2^x has a cyclicity of 4
Therefore, Rem(200/4) = 0

Rem(2^0/7) =1

Am i missing something here?

OA is D

I think you are getting confused between cyclicity of last digit and cyclicity of remainders.

2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64

If you see, the last digits are 2, 4, 8, 6 i.e. cyclicity of 4.

On the other hand,

2^1/7 Rem = 2
2^2/7 Rem = 4
2^3/7 Rem = 1
2^4/7 Rem = 2
2^5 / 7 Rem = 4
2^6/7 Rem = 1

Here the cyclicity is 3.
$$2^{198}$$ will give a remainder of 1. $$2^{200}$$ gives a remainder of 4.

Or, you can easily use binomial theorem here.
$$\frac{2^{200}}{7} = 2*2*\frac{2^{198}}{7} = 4*\frac{8^{66}}{7} = 4*\frac{(7 + 1)^{66}}{7}$$

Remainder must be 4. (Check out this link: http://www.veritasprep.com/blog/2011/05 ... ek-in-you/)
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16 Oct 2012, 20:18
4
1
$$2^{200} = (2^{5})^{40}$$

32 =28 +4

$$= (M7+4)^{40}$$ ; M7 is a multiple of 7

= M7+4

so the remainder is 4.

Hope it helps.
##### General Discussion
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Joined: 24 May 2012
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16 Oct 2012, 20:11
2
2
Actually the cyclicity is 3:

(2^0)/7 = 0 R 1
(2^1)/7 = 0 R 2
(2^2)/7 = 0 R 4
(2^3)/7 = 1 R 1
(2^4)/7 = 2 R 2
(2^5)/7 = 4 R 4
...

The pattern is such that the remainder is 4 for every third one and we know there are 201 numbers between 0 and 200. Since 201 is a multiple of 3, D is in fact correct.
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16 Oct 2012, 20:19
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17 Oct 2012, 00:17
1
mindmind wrote:
$$2^{200} = (2^{5})^{40}$$

32 =28 +4

$$= (M7+4)^{40}$$ ; M7 is a multiple of 7

= M7+4

so the remainder is 4.

Hope it helps.

$$(M7+4)^{40}=M7+4^{40}$$ then you have to find the remainder of $$4^{40}$$ when divided by 7.

Instead of taking $$32 = 28 +4$$, look for a power of 2 which gives a remainder of 1 when divided by 7.
The smallest one is $$8 = 7 + 1$$.

Therefore, $$2^{200}=(2^3)^{66}\cdot{2^2}=8^{66}\cdot{4}=(7+1)^{66}\cdot{4}=(M7+1)\cdot4=M7+4$$.
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17 Oct 2012, 00:29
EvaJager wrote:
mindmind wrote:
$$2^{200} = (2^{5})^{40}$$

32 =28 +4

$$= (M7+4)^{40}$$ ; M7 is a multiple of 7

= M7+4

so the remainder is 4.

Hope it helps.

$$(M7+4)^{40}=M7+4^{40}$$ then you have to find the remainder of $$4^{40}$$ when divided by 7.

Instead of taking $$32 = 28 +4$$, look for a power of 2 which gives a remainder of 1 when divided by 7.
The smallest one is $$8 = 7 + 1$$.

Therefore, $$2^{200}=(2^3)^{66}\cdot{2^2}=8^{66}\cdot{4}=(7+1)^{66}\cdot{4}=(M7+1)\cdot4=M7+4$$.

The remainder of 4^40 would be same as that of 4 .
Is my approach wrong? Can you provide similar examples, where it is not.
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Updated on: 17 Oct 2012, 02:48
1
mindmind wrote:
EvaJager wrote:
mindmind wrote:
$$2^{200} = (2^{5})^{40}$$

32 =28 +4

$$= (M7+4)^{40}$$ ; M7 is a multiple of 7

= M7+4

so the remainder is 4.

Hope it helps.

$$(M7+4)^{40}=M7+4^{40}$$ then you have to find the remainder of $$4^{40}$$ when divided by 7.

Instead of taking $$32 = 28 +4$$, look for a power of 2 which gives a remainder of 1 when divided by 7.
The smallest one is $$8 = 7 + 1$$.

Therefore, $$2^{200}=(2^3)^{66}\cdot{2^2}=8^{66}\cdot{4}=(7+1)^{66}\cdot{4}=(M7+1)\cdot4=M7+4$$.

The remainder of 4^40 would be same as that of 4 .
Is my approach wrong? Can you provide similar examples, where it is not.

In this case, $$4^{40}$$ is a $$M7+4$$ : $$\,\,4^{40}=2^{80}$$ and $$80=M3+2$$ (the cycle is 3, because $$2^{3}=8=M7+1$$).
$$4^{40}=2^{80}=(2^3)^{26}\cdot{2^2}=(M7+1)\cdot{4}=M7+4$$.
Or - $$4^3=64=M7+1$$, therefore $$4^{40}=(4^3)^{13}\cdot{4}=(M7+1)^{13}\cdot{4}=(M7+1)\cdot{4}=M7+4$$.

Consider for example $$5^{40}$$: $$5=7\cdot{0}+5=M7+5$$ and $$5^3=125=126-1=7\cdot{18}-1=M7-1=M7+6$$.
$$5^{40}=(5^3)^{13}\cdot{5}=(M7-1)^{13}\cdot{5}=(M7-1)\cdot{5}=M7-5=M7+2$$ and not $$M7+5$$.
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Originally posted by EvaJager on 17 Oct 2012, 02:29.
Last edited by EvaJager on 17 Oct 2012, 02:48, edited 1 time in total.
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17 Oct 2012, 02:43

The remainder of 4^40 would be same as that of 4 .
Is my approach wrong? Can you provide similar examples, where it is not.[/quote]

In this case, $$4^{40}$$ is a $$M7+4$$ : $$\,\,4^{40}=2^{80}$$ and $$80=M3+2$$ (the cycle is 3, because $$2^{3}=8=M7+1$$).
$$4^{40}=2^{80}=(2^3)^{26}\cdot{2^2}=(M7+1)\cdot{4}=M7+4$$.

Consider for example $$5^{40}$$: $$5=7\cdot{0}+5=M7+5$$ and $$5^3=125=126-1=7\cdot{18}-1=M7-1=M7+6$$.
$$5^{40}=(5^3)^{13}\cdot{5}=(M7-1)^{13}\cdot{5}=(M7-1)\cdot{5}=M7-5=M7+2$$ and not $$M7+5$$.[/quote]

Yes, Agreed
So I should consider : Something near to Remainder 1

Any Trend where, (number or prime )^n divided by (other no, prime) gives the same remainder as the original no. or prime
eg : 4^40 divided by 7 give a remainder of 4.
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17 Oct 2012, 03:01
mindmind wrote:

The remainder of 4^40 would be same as that of 4 .
Is my approach wrong? Can you provide similar examples, where it is not.

In this case, $$4^{40}$$ is a $$M7+4$$ : $$\,\,4^{40}=2^{80}$$ and $$80=M3+2$$ (the cycle is 3, because $$2^{3}=8=M7+1$$).
$$4^{40}=2^{80}=(2^3)^{26}\cdot{2^2}=(M7+1)\cdot{4}=M7+4$$.

Consider for example $$5^{40}$$: $$5=7\cdot{0}+5=M7+5$$ and $$5^3=125=126-1=7\cdot{18}-1=M7-1=M7+6$$.
$$5^{40}=(5^3)^{13}\cdot{5}=(M7-1)^{13}\cdot{5}=(M7-1)\cdot{5}=M7-5=M7+2$$ and not $$M7+5$$.[/quote]

Yes, Agreed
So I should consider : Something near to Remainder 1

Any Trend where, (number or prime )^n divided by (other no, prime) gives the same remainder as the original no. or prime
eg : 4^40 divided by 7 give a remainder of 4.[/quote]

I am not sure what you mean here:
Any Trend where, (number or prime )^n divided by (other no, prime) gives the same remainder as the original no. or prime
eg : 4^40 divided by 7 give a remainder of 4.

$$3^4=81=11\cdot{7}+4$$. $$7$$ is prime, but the remainder is neither $$3$$, nor a prime.
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17 Oct 2012, 03:09
EvaJager wrote:
mindmind wrote:

The remainder of 4^40 would be same as that of 4 .
Is my approach wrong? Can you provide similar examples, where it is not.

In this case, $$4^{40}$$ is a $$M7+4$$ : $$\,\,4^{40}=2^{80}$$ and $$80=M3+2$$ (the cycle is 3, because $$2^{3}=8=M7+1$$).
$$4^{40}=2^{80}=(2^3)^{26}\cdot{2^2}=(M7+1)\cdot{4}=M7+4$$.

Consider for example $$5^{40}$$: $$5=7\cdot{0}+5=M7+5$$ and $$5^3=125=126-1=7\cdot{18}-1=M7-1=M7+6$$.
$$5^{40}=(5^3)^{13}\cdot{5}=(M7-1)^{13}\cdot{5}=(M7-1)\cdot{5}=M7-5=M7+2$$ and not $$M7+5$$.

Yes, Agreed
So I should consider : Something near to Remainder 1

Any Trend where, (number or prime )^n divided by (other no, prime) gives the same remainder as the original no. or prime
eg : 4^40 divided by 7 give a remainder of 4.[/quote]

I am not sure what you mean here:
Any Trend where, (number or prime )^n divided by (other no, prime) gives the same remainder as the original no. or prime
eg : 4^40 divided by 7 give a remainder of 4.

$$3^4=81=11\cdot{7}+4$$. $$7$$ is prime, but the remainder is neither $$3$$, nor a prime.[/quote]

Yes.. I was referring to 3.. and not the remainder 4..
But I got your point.. thanks..
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Re: What is the remainder when you divide 2^200 by 7?  [#permalink]

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26 Oct 2012, 10:34
Another approach is the Theorem of Remainder
2^200 = 4*(2^3)^66;7 = 2^3 -1
Theorem of Remainder of f(X)/X-a is f(a)
2^200 = 4*f(X)/X-a X=2^3 a = 1 ==> Remainder of f(X)/X-a = 1
Remainder of 2^200 divided by7 is 4*1 = 4
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Re: What is the remainder when you divide 2^200 by 7?  [#permalink]

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08 Aug 2013, 22:39
Easiest and the smallest possible solution ever:

Keep your mind open while dealing with such question.It is not that you have to be math pro for that.

REM(2^200/7) [ REM(x/y) means remainder when x is divided by y]

We know that : Rem when 8 is divided by 7 is '1'.

Also by powers of 2 we can reach to 8.Using this concept:

[ (2^3)^198 * 2^2] /7

[ (8)^198 * 2^2] /7

Since REM(8/7) =1

We are left with REM(4/7) = 4
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09 Aug 2013, 01:56
1
VeritasPrepKarishma wrote:
g3kr wrote:

What is the remainder when you divide 2^200 by 7?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

my approach :

2^x has a cyclicity of 4
Therefore, Rem(200/4) = 0

Rem(2^0/7) =1

Am i missing something here?

OA is D

I think you are getting confused between cyclicity of last digit and cyclicity of remainders.

2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64

If you see, the last digits are 2, 4, 8, 6 i.e. cyclicity of 4.

On the other hand,

2^1/7 Rem = 2
2^2/7 Rem = 4
2^3/7 Rem = 1
2^4/7 Rem = 2
2^5 / 7 Rem = 4
2^6/7 Rem = 1

Here the cyclicity is 3.
$$2^{198}$$ will give a remainder of 1. $$2^{200}$$ gives a remainder of 4.

Or, you can easily use binomial theorem here.
$$\frac{2^{200}}{7} = 2*2*\frac{2^{198}}{7} = 4*\frac{8^{66}}{7} = 4*\frac{(7 + 1)^{66}}{7}$$

Remainder must be 4. (Check out this link: http://www.veritasprep.com/blog/2011/05 ... ek-in-you/)

...
2^200 = (2^3)^66 × 2^2 = (7+1)^66 × 4
1^66 × 4 = 4 (Answer)

The taste of your own medicine ???????
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Re: What is the remainder when you divide 2^200 by 7?  [#permalink]

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21 Mar 2016, 08:37
We can use the binomial approach here => 4*2^198 => 4* (7P+1) => 7Q+4 => remainder => 4
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Re: What is the remainder when you divide 2^200 by 7?  [#permalink]

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24 May 2020, 08:00
g3kr wrote:

What is the remainder when you divide 2^200 by 7?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

my approach :

2^x has a cyclicity of 4
Therefore, Rem(200/4) = 0

Rem(2^0/7) =1

Am i missing something here?

OA is D

I think you are getting confused between cyclicity of last digit and cyclicity of remainders.

2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64

If you see, the last digits are 2, 4, 8, 6 i.e. cyclicity of 4.

On the other hand,

2^1/7 Rem = 2
2^2/7 Rem = 4
2^3/7 Rem = 1
2^4/7 Rem = 2
2^5 / 7 Rem = 4
2^6/7 Rem = 1

Here the cyclicity is 3.
$$2^{198}$$ will give a remainder of 1. $$2^{200}$$ gives a remainder of 4.

Or, you can easily use binomial theorem here.
$$\frac{2^{200}}{7} = 2*2*\frac{2^{198}}{7} = 4*\frac{8^{66}}{7} = 4*\frac{(7 + 1)^{66}}{7}$$

Remainder must be 4. (Check out this link: http://www.veritasprep.com/blog/2011/05 ... ek-in-you/)

VeritasKarishma : why this question can't be solved with cyclicity approach, here is similar question thread i am copying, and is solved by cylicity. why e need to bring remainder cylicity.

https://gmatclub.com/forum/what-is-the- ... l#p2066643
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Re: What is the remainder when you divide 2^200 by 7?  [#permalink]

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25 May 2020, 03:07
1
rishab0507 wrote:
g3kr wrote:

What is the remainder when you divide 2^200 by 7?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

my approach :

2^x has a cyclicity of 4
Therefore, Rem(200/4) = 0

Rem(2^0/7) =1

Am i missing something here?

OA is D

I think you are getting confused between cyclicity of last digit and cyclicity of remainders.

2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64

If you see, the last digits are 2, 4, 8, 6 i.e. cyclicity of 4.

On the other hand,

2^1/7 Rem = 2
2^2/7 Rem = 4
2^3/7 Rem = 1
2^4/7 Rem = 2
2^5 / 7 Rem = 4
2^6/7 Rem = 1

Here the cyclicity is 3.
$$2^{198}$$ will give a remainder of 1. $$2^{200}$$ gives a remainder of 4.

Or, you can easily use binomial theorem here.
$$\frac{2^{200}}{7} = 2*2*\frac{2^{198}}{7} = 4*\frac{8^{66}}{7} = 4*\frac{(7 + 1)^{66}}{7}$$

Remainder must be 4. (Check out this link: http://www.veritasprep.com/blog/2011/05 ... ek-in-you/)

VeritasKarishma : why this question can't be solved with cyclicity approach, here is similar question thread i am copying, and is solved by cylicity. why e need to bring remainder cylicity.

https://gmatclub.com/forum/what-is-the- ... l#p2066643

Cyclicity tells you the units digit of exponential expressions. Now think about this - if you know the units digit of a number, can you say what the remainder is upon division by say 7?

e.g. What is the remainder when 792947 is divided by 7? Would you say the remainder here is 0? It is not.
The remainder depends on what the actual number is, not just the units digit.

Only in case of division by 2, 5 or 10 does the units digit give us the remainder.

Check out these two posts:
https://www.veritasprep.com/blog/2015/1 ... questions/
https://www.veritasprep.com/blog/2015/1 ... ns-part-2/
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Re: What is the remainder when you divide 2^200 by 7?  [#permalink]

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25 May 2020, 11:12
rishab0507 wrote:
g3kr wrote:

What is the remainder when you divide 2^200 by 7?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

my approach :

2^x has a cyclicity of 4
Therefore, Rem(200/4) = 0

Rem(2^0/7) =1

Am i missing something here?

OA is D

I think you are getting confused between cyclicity of last digit and cyclicity of remainders.

2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64

If you see, the last digits are 2, 4, 8, 6 i.e. cyclicity of 4.

On the other hand,

2^1/7 Rem = 2
2^2/7 Rem = 4
2^3/7 Rem = 1
2^4/7 Rem = 2
2^5 / 7 Rem = 4
2^6/7 Rem = 1

Here the cyclicity is 3.
$$2^{198}$$ will give a remainder of 1. $$2^{200}$$ gives a remainder of 4.

Or, you can easily use binomial theorem here.
$$\frac{2^{200}}{7} = 2*2*\frac{2^{198}}{7} = 4*\frac{8^{66}}{7} = 4*\frac{(7 + 1)^{66}}{7}$$

Remainder must be 4. (Check out this link: http://www.veritasprep.com/blog/2011/05 ... ek-in-you/)

VeritasKarishma : why this question can't be solved with cyclicity approach, here is similar question thread i am copying, and is solved by cylicity. why e need to bring remainder cylicity.

https://gmatclub.com/forum/what-is-the- ... l#p2066643

Cyclicity tells you the units digit of exponential expressions. Now think about this - if you know the units digit of a number, can you say what the remainder is upon division by say 7?

e.g. What is the remainder when 792947 is divided by 7? Would you say the remainder here is 0? It is not.
The remainder depends on what the actual number is, not just the units digit.

Only in case of division by 2, 5 or 10 does the units digit give us the remainder.

Check out these two posts:
https://www.veritasprep.com/blog/2015/1 ... questions/
https://www.veritasprep.com/blog/2015/1 ... ns-part-2/

thanks, It was indeed a good learning. I never paid attention i was solving only 2,5, 10 divisor problems .I think for other divisors i need to follow split method, not cyclicity to find remainders
Re: What is the remainder when you divide 2^200 by 7?   [#permalink] 25 May 2020, 11:12