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jasen

Find the number (my version)

Question

based on this @bonanova puzzle, I create another similar puzzle

You are given the following ten statements and are asked to determine a particular number.

  1. At least one of statements 7 and 8 is true.
  2. This either is the first true or the first false statement.
  3. The number is a prime number.
  4. The first true statement multiplied by the last false statement divides the number.
  5. The number of divisors of the number is greater than the sum of the numbers of the true statements.
  6. The number has exactly 4 prime divisors.
  7. The number is bigger than 1000.
  8. The numbers of true statements do not equal the numbers of false statement
  9. One of the divisors is a cube number bigger than 1.
  10. There are 3 consecutive False statements and 3 consecutive True statements.

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Spoiler

The number is 840

Start with number 2. Must be True. That makes number 1 False. Which makes 7 and 8 False. With 7 False we know the number is <=1000. With 8 False we know there are exactly 5 True and 5 False statements

So Far we have 1)F 2)T 3) 4) 5) 6) 7)F 8)F 9) 10)

Now assume number 3 is true. If the number is prime, then 4, 5, 6, and 9 must be false. That would mean that there are 7 false statements which contradicts the conclusion about number 8. Therefore 3 must be False.

Now we have 1)F 2)T 3)F 4) 5) 6) 7)F 8)F 9) 10)

Next we look at statement 10. Assume it is True. This makes 9 False and 4,5,6 True. The number would be defined as <1000 (statement 7), has 18 as divisor (statement 4), has >27 divisors(statement 5 {2+4+5+6+10}), has 4 prime divisors (statement 6), and divisor is not a cube (statement 9). The only number divisible by 18 that has more than 27 factors is 720. This is divisible by 8 and therefore can not be true. So statement 10 must be False.

This results in the statements as follows 1)F 2)T 3)F 4)T 5)T 6)T 7)F 8)F 9)T 10)F

So the number must be <1000, divisible by 20, divisible by a cube, have 4 prime divisors, and have >26 divisors. The only number that fits this is 840.

 

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