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The remainder when 4^1000 is divided by 7 equals:

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The remainder when 4^1000 is divided by 7 equals:  [#permalink]

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New post Updated on: 14 Oct 2018, 06:52
2
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A
B
C
D
E

Difficulty:

  65% (hard)

Question Stats:

57% (01:40) correct 43% (00:53) wrong based on 14 sessions

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The remainder when 4^1000 is divided by 7 equals:

A) 3
B) 2
C) 4
D) none of these

Originally posted by asifahmed500 on 14 Oct 2018, 04:59.
Last edited by generis on 14 Oct 2018, 06:52, edited 2 times in total.
Edited the OA
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Re: The remainder when 4^1000 is divided by 7 equals:  [#permalink]

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New post 14 Oct 2018, 05:09
asifahmed500 wrote:
The remainder when 4^1000 is divided by 7 equals:

1) 3
2) 2
3) 4
4) none of these

Posted from my mobile device


4^1000 = 4 ( 4^999 ) = 4 ( 64^333 ) = 4 ( ( 63 + 1 ) ^ 333 )

All the terms in the expansion of ( 63 + 1 )^333 shall contain 63 except the last term i.e. 1
So 4 shall be the remainder.

Please update the OA.
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Re: The remainder when 4^1000 is divided by 7 equals:  [#permalink]

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New post 14 Oct 2018, 05:14
Is it possible to solve this problem with the help of cyclicity approach

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Re: The remainder when 4^1000 is divided by 7 equals:  [#permalink]

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New post 14 Oct 2018, 05:32
asifahmed500 wrote:
The remainder when 4^1000 is divided by 7 equals:

1) 3
2) 2
3) 4
4) none of these

Posted from my mobile device


OA: C

Remainder
\(\frac{4^1}{7} = \frac{4}{7} = 4\)
\(\frac{4^2}{7} = \frac{16}{7} = 2\)
\(\frac{4^3}{7} = \frac{64}{7} = 1\)
\(\frac{4^4}{7} = \frac{256}{7} = 4\)
\(\frac{4^5}{7} = \frac{1024}{7} = 2\)

Remainder when \(4^n\) divided by 7 has a cyclicity of 3

\(4^{1000} =4^{3*333+1}\)
So remainder will be 4
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Re: The remainder when 4^1000 is divided by 7 equals:  [#permalink]

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New post 14 Oct 2018, 05:47
\(4^{1000}\)
=\((4^3)^{333}\) x4
= \(64^{333}\) x 4
remainder when \(4^{1000}\) is divided by 7 = remainder when \(64^{333}\) x 4 is divided by 7
Remainder \((64^{1000})\)/7 X Remainder \(\frac{4}{7}\)
\(1^{333}\) X 4
= 4
Remainder = 4

asifahmed500 wrote:
The remainder when 4^1000 is divided by 7 equals:

1) 3
2) 2
3) 4
4) none of these

Posted from my mobile device

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Re: The remainder when 4^1000 is divided by 7 equals:   [#permalink] 14 Oct 2018, 05:47
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The remainder when 4^1000 is divided by 7 equals:

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