What is the remainder when 2^2006 is divided by 7?

Question

What is the remainder when  2^{2006}  is divided by 7?

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

Are You Up For the Challenge: 700 Level Questions

EgmatQuantExpert · Accepted Answer

Solution

To find  
We need to determine
 • The remainder, when  2^{2006}  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, 2006 = 3 x k + 2
Hence, Rem 2^{2006}/7 ] = Rem 2^{3k+2}/7 ] = Rem 2^{3k}/7 ] * Rem 2^2/7 ] = 1 x 4 = 4

Thus, option B is the correct answer.

Correct Answer: Option B

lacktutor · Answer

What is the remainder when  2^{2006}  is divided by  7 ?

2^{2006}= 4*2^{2004}= 4*2^{3*668}= 4*8^{668}= 4*(7+1)^{668}

-->  4(....+1)^{668}   --> The remainder will be  4 .

The answer is B .

GMATWhizTeam · Answer

Solution  
 • We know that when ( 2^3 = 8 ) is divided by 7 remainder is 1.
 o So, when  2^{3n}  will divided by 7 remainder will be 1, where n is a positive integer.  
• So, lets express the power of  2^{2006}  (i.e. 2006 ) as multiple of 3.
•   _{remainder} =  _{remainder} = 4* _{remainder} = 4*1 = 4   
Thus, the correct answer is  Option B

DaniyalAlwani · Answer

EgmatQuantExpert

I'm really struggling in the method you have used? Can you please explain me this concept in detail?

ScottTargetTestPrep · Answer

We see that 2^2006 = (2^3)^668 * 2^2 = 8^668 * 4. Since 8 has a remainder of 1 when divided by 7, 8^668 also has a remainder of 1 when divided by 7 (notice that 1^668 = 1). Therefore, the remainder when 2^2006 = 8^668 * 4 is divided by 7 is 1 * 4 = 4.

Answer: B

omavsp · Answer

2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
this pattern goes on.

now 2006/4 = r2
our digit unit with r2 = 2^2 = 4
ans 4/7

BrushMyQuant · Answer

We need to find the the remainder when  2^{2006}  is divided by 7 ?

Let's solve the problem using two Methods:

Method 1: Cyclicity of Remainder of power of 2 by 7

To solve this problem we need to find the cycle of remainder of power of 2 when divided by 7

Remainder of  2^1  (=2) by 7 = 2
Remainder of  2^2  (=4) by 7 = 4 
Remainder of  2^3  (=8) by 7 = 1
Remainder of  2^4  (=16) by 7 =   2   
Remainder of  2^5  (=32) by 7 =   4  
Remainder of  2^6  (=64) by 7 =   1

=> Cycle is 3

Remainder of 2006 by 3 = 2
=> Remainder of  2^{2006}  by 7 = Remainder of 2^2 by 7 =4

Method 2: Binomial Theorem

2^{2006}  =  2^{2004+2}  = 2^2004 * 2^2 = 2^(3*668) * 4 = 8^668 * 4

Remainder of  2^{2006}  by 7 = Remainder of 8^668 by 7 * Remainder of 4 by 7 = (Remainder of 8^668) * 4

Remainder of 8^668 = Remainder of (7 + 1)^668 by 7
 
 Now, if we use Binomial Theorem to expand this then all the terms except the last term will be a multiple of 7
=> All terms except the last term will give remainder of 0 when divided by 7 
=> Problem is reduced to what is the remainder when the last term (i.e. 668C668 * 7^0 * 1^668) is divided by 7
=> Remainder of 1 * 1 * 1 is divided by 7
=> Remainder of 1 divided by 7 = 1
=> Remainder of  2^{2006}  by 7 = (Remainder of 8^668) * 4 by 7 = 1 * 4 = 4

So,  Answer will be B 
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

What is the remainder when 2^2006 is divided by 7?

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

What is the remainder when 2^2006 is divided by 7?

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards