What is the remainder when 2^2004 is divided by 7?

Question

What is the remainder when  2^{2004}  is divided by 7?

A. 5
B. 4
C. 3
D. 2
E. 1

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chetan2u · Accepted Answer

You can do it in two ways

(I) Pattern
If GMAT asks you a question of tis kind, there has to be some pattern
 2^0  divided by 7 gives 1 as remainder
 2^1  divided by 7 gives 2 as remainder
 2^2  divided by 7 gives 4 as remainder
 2^3  divided by 7 gives 1 as remainder
 2^4  divided by 7 gives 2 as remainder
So pattern is 2, 4, 1, 2, 4, 1... A multiple of 3 as the power will give 1 as remainder..
2004=3*668...so a multiple of 3, hence remainder is 1

(II) Binomial Theorem
 2^{2004}=(2^3)^{668}=8^{668}=(7+1)^{668}=7^{668}1^0+7^{667}1^1+....7^11^{667}+1^{668} 
Hence all terms except  1^{668}  are divisible by 7, so remainder =1

E

shameekv1989 · Answer

Check for cyclicity 
 \frac{2}{4}  - remainder - 2 (Power = 1)
 \frac{4}{7}  - remainder - 4 (Power = 2)
 \frac{8}{7}  - remainder -1 (Power = 3) 
 \frac{16}{7}  - remainder - 2 (Power = 1+3) 
 \frac{32 }{7}  - remainder - 4 (Power = 2+3) 
 \frac{64}{7}  - remainder - 1 (Power = multiple of 3)

Therefore a cyclicity of 3

Check for the remainder of Power with cyclicity =>  \frac{2004}{3} = 0 ; hence the powers of 3 i.e. remainder = 1

Answer - E

lacktutor · Answer

What is the remainder when  2^{2004}  is divided by  7 ?

2^{2004}= 2^{3*668}= 8^{668}= (7+1)^{668}

--> the remainder will be  1 .

The answer is E.

EgmatQuantExpert · Answer

Solution

To find  
We need to determine
 • The remainder, when  2^{2004}  is divided by 7

Approach and Working out  
We know,  2^3  divided by 7 gives a remainder 1.
 • Hence, we can also say  2^{3k}  divided by 7 also gives a remainder 1.

Now, 2004 = 3 x k
Hence, Rem 2^{2004}/7 ] = Rem 2^{3k}/7 ] = 1

Thus, option E is the correct answer.

Correct Answer: Option E

curatelycurious · Answer

I did it in this way, correct me if i am wrong pls: 2^(4*501)/7, 2^4 gives remainder as 6 which -1 when divided by 7 and 2^501 gives remainder as 1 so i chose 1

What is the remainder when 2^2004 is divided by 7?

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GMAT Problem Solving (PS) Questions

What is the remainder when 2^2004 is divided by 7?

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