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 numbers of the last true and
the first true statement divides the number, that is to be
found.
5. The sum of the numbers of the true statements is the
number, that is to be found.
6. This is not the last true statement.
7. The number of each true statement divides the number,
that is to be found.
8. The number 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.
Find the minimal possible number?
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Answer Posted / guest
The numebr is 420.
If statement 6 is false, it creates a paradox. Hence,
Statement 6 must be true.
Consider Statement 2:
? If it is true, it must be the first true statement.
Otherwise, it creates a paradox.
? If it is false, it must be the second false statement.
Otherwise, it creates a paradox.
In both the cases, Statement 1 is false.
As Statement 1 is false, Statement 9 and Statement 10 both
are false i.e. there are three consecutive true statements.
1 2 3 4 5 6 7 8 9 10
False - - - - True - - False False
Let\'s assume that Statement 3 is false i.e. there are no
three consecutive false statements. It means that Statement
2 and Statement 8 must be true, else there will be three
consecutive false statements.
1 2 3 4 5 6 7 8 9 10
False True False - - True - True False False
Also, atleast two of Statements 4, 5 and 7 must be true as
there are three consecutive true statements.
According to Statement 8, the number that is to be found is
the percentage of true statements. Hence, number is either
50 or 60. Now if Statement 7 is true, then the number of
each true statement divides the number, that is to be found.
But 7 and 8 do not divide either 50 or 60. Hence, Statement
7 is false which means that Statement 4 and 5 are true. But
Statement 5 contradicts the Statement 8. Hence, our
assumption that Statement 3 is false is wrong and Statement
3 is true i.e. there are 3 consecutive false statements
which means that Statement 8 is false as there is no other
possibilities of 3 consecutive false statements.
Also, Statement 7 is true as Statement 6 is not the last
true statement.
1 2 3 4 5 6 7 8 9 10
False - True - - True True False False False
According to Statement 7, the number of each true statement
divides the number, that is to be found. And according to
Statement 5, the sum of the numbers of the true statements
is the number, that is to be found. For all possible
combinations Statement 5 is false.
There 3 consecutive true statements. Hence, Statement 2 and
Statement 4 are true.
1 2 3 4 5 6 7 8 9 10
False True True True False True True False False False
Now, the conditions for the number to be found are:
1. The numebr is divisible by 5 (Statement 4)
2. The numebr is divisible by 2, 3, 4, 6, 7 (Statement 7)
3. The number of divisors of the number, that is to be
found, (apart from 1 and itself) is not greater than the sum
of the numbers of the true statements. (Statement 9)
The minimum possible number is 420.
The divisors of 420, apart from 1 and itself are 2, 3, 4, 5,
6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84,
105, 140, 210. There are total of 22 divisors. Also, the sum
of the numbers of the true statements is 22 (2+3+4+6+7=22),
which satisfies the third condition.
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