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The Ten Statements II (Posted on 2014-05-08) Difficulty: 3 of 5
There are 10 statements written on a piece of paper:
  1. At least one of statements 9 and 10 is true.
  2. This either is the first true or the first false statement.
  3. There are three consecutive statements, which are false.
  4. The difference between the serial numbers of the last true and the first true statement divides the positive integer that is to be found.
  5. The sum of the numbers of the true statements is the positive integer that is to be found.
  6. This is not the last true statement.
  7. The number of each true statement divides the positive integer that is to be found.
  8. The positive integer that is to be found is the percentage of true statements.
  9. The number of divisors of the number that is to be found, (apart from 1 and itself) is greater than the sum of the numbers of the true statements.
  10. There are no three consecutive true statements.
What is the smallest possible value of the positive integer that is to be found?

See The Solution Submitted by K Sengupta    
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Solution Solution | Comment 1 of 7

Another way of rewording statement 2 is "This statement has a different truth value than Statement 1."  Thus, if Statement 2 is true, then Statement 1 is false; if Statement 2 is false, then Statement 1 is false.  Either way, Statement 1 is false.

From this we know that Statements 9 and 10 are both false.

From Statement 10 we know that there are three consecutive true statements. 

Statement 6 must be true. 

From this we know that 7 and/or 8 is true.  However, we see they can't both be true, as 8 implies that the integer we seek is less than or equal to 70 and 7 would imply that that the integer is at least LCM(6,7,8) = 168.  So exactly one of 7 or 8 is true.

Also, 5 and 8 cannot both be true.

Therefore, if 8 is true, then 5 and 7 are false.  Since we need three consecutive true statements, this would make 2, 3, and 4 true.  But then there wouldn't be three consecutive false statements, which violates statement 3.  Thus 8 is false, and 7 is true.

We know from 7 that the integer we seek is at least LCM(6,7) = 42.  Then there is no way for 5 to be true, so 5 is false.  And since we require at least 3 consecutive true statements, then 2, 3, and 4 are all true.

From 4 we know the integer we seek is divisible by 5, and from 7 we know that it's divisible by 2, 3, 4, 6, and 7.  Thus the smallest possible value for the integer is LCM(2,3,4,5,6,7) = 420

  Posted by tomarken on 2014-05-08 16:48:30
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