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

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Re: What is the remainder when 2^2006 is divided by 7? [#permalink]
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Bunuel wrote:
What is the remainder when $$2^{2006}$$ is divided by 7?

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

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.
• $$[\frac{2^{2006}}{7}]_{remainder} = [\frac{(2^{2004}*2^2)}{7}]_{remainder} = 4*[\frac{(2^{2004})}{7}]_{remainder} = 4*1 = 4$$
Thus, the correct answer is Option B
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Re: What is the remainder when 2^2006 is divided by 7? [#permalink]
EgmatQuantExpert

I'm really struggling in the method you have used? Can you please explain me this concept in detail?
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Re: What is the remainder when 2^2006 is divided by 7? [#permalink]
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Bunuel wrote:
What is the remainder when $$2^{2006}$$ is divided by 7?

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

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.

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Re: What is the remainder when 2^2006 is divided by 7? [#permalink]
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
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Re: What is the remainder when 2^2006 is divided by 7? [#permalink]
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Re: What is the remainder when 2^2006 is divided by 7? [#permalink]
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