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# Find the number

## Question

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

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 false statements.
4. The difference between the numbers of the last true and the first true statement divides the number.
5. The sum of the numbers of the true statements is the number.
6. This is not the last true statement.
7. The number of each true statement divides the number.
8. The number is the percentage of true statements.
9. The number of divisors of the number (apart from 1 and itself) is greater than the sum of the numbers of the true statements.
10. No three consecutive statements are true.

What is the number ?

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Start from 2nd statement,

It must be true, because it can't be the first false statement. If it the first false statement, it become liar paradox.

So 1 False, 2 True, 9 False and 10 False.

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Doesn't statement 6 have to be true, because if it was false it wouldn't make sense?

Also, the way I read it, statement 2 could be false. That would mean it's NOT the first false statement, which makes statement 1 false. But if statement 2 is true then statement 1 is still false. So either way statement 1 is false, making 9 and 10 false, but you still don't know if 2 is true or false yet.

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Spoiler

The number is 420

If statement #2 is true, then it is the first true statement and statement #1 is false.
If statement #2 is false, then it is not the first false statement and statement #1 is false.
In either case, statement #1 is false. This means that statements #9 and #10 are also false.
If statement #6 is false, then it is definitely true. So statement #6 must be true.
If statement #3 is false, then statements #2 and #8 must be true, and if statement #8 is true, the number must be one of [40 50 60] and statements #5 (which implies that the number is less than 36) and #7 (which implies that the number is a multiple of 7) must be false. This would make statement #10 true, so statement #3 must be true.
Since statements #3 and #6 are true, statement #8 must be false (to qualify statement #3) and statement #7 must be true (to qualify statement #6).
Since statements #6 and #7 are true, the number must be a multiple of 42. Therefore statement #5 must be false.
Since statement #10 is false, statements #2 and #4 must be true.

True statements are #2, #3, #4, #6 and #7. The sum of these numbers is 22.
The number must be a multiple of 84 (statement #7) and a multiple of 5 (statement 4), therefore the number must be a multiple of 420.
The number must not have more that 22 divisors (apart from 1 and itself, statement 9). Since 420 has 22 divisors, and all greater multiples of 420 have more than 22 divisors, the number must be 420.

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