There are 10 statements written on a piece of paper:
- At least one of statements 9 and 10 is true.
- This either is the first true or the first false statement.
- There are three consecutive statements, which are false.
- The difference between the serial numbers of the last true and the first true statement divides the positive integer that is to be found.
- The sum of the numbers of the true statements is the positive integer that is to be found.
- This is not the last true statement.
- The number of each true statement divides the positive integer that is to be found.
- The positive integer that is to be found is the percentage of true statements.
- 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.
- There are no three consecutive true statements.
What is the smallest possible value of the positive integer that is to be found?
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