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?

Joe is not a knight:-
N=23l=10m+9, 0<l, 0<m<=100, 7k=10m+9

{l = 3, m = 6}: 7 does not divide (69)
{l = 13, m = 29}:  7 does not divide (299)
{l = 23, m = 52}: 7 does not divide (529)
{ l = 33, m = 75}: 7 does not divide (759)
{l = 43, m = 98}:  7 does not divide (989)

Joe is not a  liar:-
N is an even number, not divisible by 7 or 23, divisible by 19, with an odd number of divisors, not ending in 9.To have an odd number of divisors, n has to be a power; specifically all of its factors need to be raised to an even power. 19^2 is 361, but that is odd. (19*2)^2=1444, which exceeds 1000.

If Joe is a LT knave we have:
N is an even number, divisible by 7, not divisible by 19 or 23, with an odd number of divisors (see above),  ending in 9. (14x)^2 <1000, x>0 gives x={1,2}. The possibilities are 196, 784, neither of which ends in 9.

If Joe is a TL knave we have:
N is an odd number, not divisible by 7 but divisible by 19 and 23, with an even number of divisors, not ending in 9. 19*23=437 is the solution, since three times this number exceeds 1000.

Posted by broll on 2011-04-28 03:18:05