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Director  Joined: 25 Apr 2012
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A repunit is a positive integer that contains only the digit  [#permalink]

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Question Stats: 65% (01:30) correct 35% (01:30) wrong based on 368 sessions

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A repunit is a positive integer that contains only the digit 1. If the integer r is a repunit, is r prime?

(1) The number of digits in r is a multiple of 3.
(2) 0 < r < 1200

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“If you can't fly then run, if you can't run then walk, if you can't walk then crawl, but whatever you do you have to keep moving forward.”

Originally posted by WoundedTiger on 22 Jul 2014, 02:03.
Last edited by Bunuel on 22 Jul 2014, 02:52, edited 1 time in total.
Edited the question.
Math Expert V
Joined: 02 Sep 2009
Posts: 58434
Re: A repunit is a positive integer that contains only the digit  [#permalink]

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A repunit is a positive integer that contains only the digit 1. If the integer r is a repunit, is r prime?

As defined repunit numbers are 1, 11, 111, 1111, ....

(1) The number of digits in r is a multiple of 3. This implies that the sum of the digits is 3*1 (for 111), 6*1 (for 111,111), 9*1 (for 111,111,111), ... So in any case the sum of the digits is a multiple of 3, hence the number is divisible by 3, which means that it cannot be a prime. Sufficient.

(2) 0 < r < 1200. If r = 11, then it's a rpime but if r = 111, then it's not a prime. Not sufficient.

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A repunit is a positive integer that contains only the digit  [#permalink]

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Bunuel wrote:
A repunit is a positive integer that contains only the digit 1. If the integer r is a repunit, is r prime?

As defined repunit numbers are 1, 11, 111, 1111, ....

(1) The number of digits in r is a multiple of 3. This implies that the sum of the digits is 3*1 (for 111), 6*1 (for 111,111), 9*1 (for 111,111,111), ... So in any case the sum of the digits is a multiple of 3, hence the number is divisible by 3, which means that it cannot be a prime. Sufficient.

(2) 0 < r < 1200. If r = 11, then it's a rpime but if r = 111, then it's not a prime. Not sufficient.

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Re: A repunit is a positive integer that contains only the digit  [#permalink]

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WoundedTiger wrote:
A repunit is a positive integer that contains only the digit 1. If the integer r is a repunit, is r prime?

(1) The number of digits in r is a multiple of 3.
(2) 0 < r < 1200

Target question: Is r prime?

Given: Integer r is a repunit
So, some possible values of r include: 1, 11, 111, 1111, 111111111111111111, etc

Statement 1: The number of digits in r is a multiple of 3.
So, r could equal 111 or 111,111 or 111,111,111 or 111,111,111,111 etc
We know that, if the sum of the digits in a number is divisible by 3, then that number is also divisible by 3
For example, since the digits in 111 add to 3 (and 3 is divisible by 3), we know that 111 is divisible by 3, which means 111 is NOT prime.
Likewise, since the digits in 111,111 add to 6 (and 6 is divisible by 3), we know that 111,111 is divisible by 3, which means 111,111 is NOT prime.
And, since the digits in 111,111,111 add to 9 (and 9 is divisible by 3), we know that 111,111,111 is divisible by 3, which means 111,111,111 is NOT prime.
As we can see, if the number of digits in r is a multiple of 3, then the sum of r's digits will be divisible by 3.
So, the answer to the target question is NO, r is NOT prime
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: 0 < r < 1200
There are 4 possible values of r: 1 or 11 or 111 or 1111
Case a: If r = 11, then r IS prime
Case b:If r = 111, then r is NOT prime
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Cheers,
Brent
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Re: A repunit is a positive integer that contains only the digit  [#permalink]

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_________________ Re: A repunit is a positive integer that contains only the digit   [#permalink] 16 Jan 2019, 15:45
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