Joe is either a knight, a liar, or a knave. His favorite number is an integer between 1 and 1000. He makes the following statements about his favorite number:

1. It is odd.
2. It is divisible by 7.
3. It is divisible by 23.
4. It is not divisible by 19.
5. It has an even number of divisors.
6. It ends in 9.

What type is Joe, and what is his favorite number?

Note as in "Burger Buddies", every other statement must have the same truth value.

So suppose statement 5 is false. Then statement 3 and 1 are false, so the number is even. But then it can't end in 9, so statement 6, (and thus statement 4 and statement 2) would be false. However only perfect squares have an even number of divisors, and the smallest even perfect square divisible by 19 is greater than 1000.

So statements 1,3, and 5 must be true, and so the number is divisible by 23. If it were also divisible by 7, it would have to end in 9. Noting 7*23=161, it is clear the first multiple of 161 ending in 9 would also be greater than 1000.

So statements 2, 4, and 6 must be false. The only odd multiple of 23 and 19 less than 1000 is 23*19=437, so that must be the number.

Posted by Gamer on 2011-05-12 05:16:21