The remainder when 7^84 is divided by 342 is : : GMAT Problem Solving (PS)
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# The remainder when 7^84 is divided by 342 is :

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Manager
Joined: 23 May 2013
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The remainder when 7^84 is divided by 342 is : [#permalink]

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26 Feb 2014, 09:03
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Difficulty:

15% (low)

Question Stats:

82% (02:21) correct 18% (00:00) wrong based on 17 sessions

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The remainder when 7^84 is divided by 342 is :

1. 0
2. 1
3. 49
4. 341
[Reveal] Spoiler: OA

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Magoosh GMAT Instructor
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Re: The remainder when 7^84 is divided by 342 is : [#permalink]

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26 Feb 2014, 12:38
ankur1901 wrote:
The remainder when 7^84 is divided by 342 is :

1. 0
2. 1
3. 49
4. 341

Dear ankur1901,

First, let me make perfectly clear: this is NOT a GMAT question. This is far too advanced for anything on the GMAT. This is getting into mathematical hot-shot territory.

Step one is to notice that 342 = 343 -1, and of course, 343 = 7^3, so 342 = (7^3) - 1.

Step two is to remember that (a^n - 1) is divisible by (a^m - 1) if n is divisible by m. Obviously, 84 is divisible by 3. Therefore:

[(7^84) - 1] must be divisible by [(7^3) - 1] = 342

If 342 goes evenly into [(7^84) - 1], then it must divide into (7^84) leaving a remainder of 1.

Does all this make sense?

Mike
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Mike McGarry
Magoosh Test Prep

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Re: The remainder when 7^84 is divided by 342 is : [#permalink]

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26 Feb 2014, 17:35
ankur1901 wrote:
The remainder when 7^84 is divided by 342 is :

1. 0
2. 1
3. 49
4. 341

Hi Ankur,

For this kind of problems, the best approach is to find the power of the dividend which is closest to divisor(or the multiple of the divisor)

7 cube = 343
Divisor = 342

343/342 => reminder =1
as 7^84 can be written as (7^3)^28 = the result is 1^28 =>1
Re: The remainder when 7^84 is divided by 342 is :   [#permalink] 26 Feb 2014, 17:35
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