Find the number from 10 statements

Question

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 distinct 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.

Answer 1

Our number is

840

And the truth/falsehood of the statements is

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

---

Assume statement 1 is true. Then

statement 2 can either be true, in which case it would be false because it isn't the first true statement; or it could be false, in which case it would be true because it would be the first false statement. Therefore 1 must be false.

This implies that

7 and 8 are both false. We now know the number is smaller or equal to 1000, and the number of false statements equals the number of true statements (so we must have 5 each).

Now assume 3 is true. Then, we get that

as the number is prime, 4, 6 and 9 are all false, which with 1, 7 and 8, leaves us with 6 false statements, which is impossible. So 3 is false and the number is not prime.

Continuing on,

of the remaining statements, exactly one is false, the rest are all true. If either of 4, 5 or 6 is false, this makes 10 also false. So 4, 5 and 6 are true.

Assume 9 is false. Then for 5 to be true,

the number has to have more than 2+4+5+6+10=27 divisors. But as the number is smaller than 1000, this is impossible unless it has a cube divisor greater than 1, which would mean 9 is true. So 9 has to be true.

6, 7 and 9 imply

our number is $2^3 \times 3 \times 5 \times 7 = 840$. It has 32 divisors, so for 5 to be true, 10 can't be true or the sum of the true statements would be too high. This leaves 2 as the last true statement.