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

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Intern
Joined: 11 Sep 2012
Posts: 7
The remainder when 4^1000 is divided by 7 equals:  [#permalink]

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Updated on: 14 Oct 2018, 06:52
2
00:00

Difficulty:

65% (hard)

Question Stats:

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

### HideShow timer Statistics

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
Intern
Joined: 22 Jun 2018
Posts: 28
Location: India
GMAT 1: 730 Q50 V39
GPA: 4
WE: Research (Telecommunications)
Re: The remainder when 4^1000 is divided by 7 equals:  [#permalink]

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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.

Intern
Joined: 11 Sep 2012
Posts: 7
Re: The remainder when 4^1000 is divided by 7 equals:  [#permalink]

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

Posted from my mobile device
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Joined: 22 Feb 2018
Posts: 419
Re: The remainder when 4^1000 is divided by 7 equals:  [#permalink]

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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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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

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