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

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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]
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