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A repunit is a positive integer that contains only the digit

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

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Updated on: 22 Jul 2014, 01:52
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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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Originally posted by WoundedTiger on 22 Jul 2014, 01:03.
Last edited by Bunuel on 22 Jul 2014, 01:52, edited 1 time in total.
Edited the question.
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Re: A repunit is a positive integer that contains only the digit  [#permalink]

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22 Jul 2014, 01:55
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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22 Jul 2014, 01:59
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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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03 Jan 2018, 09:41
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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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26 Apr 2020, 03:46
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Re: A repunit is a positive integer that contains only the digit   [#permalink] 26 Apr 2020, 03:46