I am a three digit number.
I am either divisible by 3 or by 5.
I am either divisible by 4 or by 6.
I am either divisible by 5 or by 7.
I am either divisible by 6 or by 8.
I am either divisible by 7 or by 9.
I am either divisible by 9 or by 11.

What am I?

(Note: "divisible" means leaving no remainder.)

(In reply to That depends on what "or" is -- computer solution by Charlie)

In logic, where "or" is used as an inclusive disjunction it is often represented as "OR".  In everyday language, "or" is usually used as an exclusive disjunction – in logic this is often represented as "XOR".
The title of the problem is given as "Either / Or", which clearly implicates the use of an exclusive disjunction and not the inclusive disjunction.

Begin with the first statement => 3 XOR 5

Adding the second statement => (3 XOR 5) AND (4 XOR 6)
But as 6 is a multiple of 3, the formula must also be =>
6 XOR (5 AND 4)

Adding the third statement to the formula gives =>
(6 AND 7) XOR (5 AND 4)

Adding the fourth statement, as 8 is a multiple of 4, gives
(6 AND 7) XOR (5 AND 8)

Adding the fifth statement would give
(6 AND 7) XOR (5 AND 8 AND 9),
but 9 is a multiple of 3, and the first statement excludes the factor 3 with 5, this leaves the solution to include the factors of 6 and 7 and exclude the factors 4, 5, 8 and 9.

Adding the sixth statement then gives => (6 AND 7 AND 11)

6*7*11 = 462

Applying any additional prime factors (as a second factor of 2 is excluded by excluding the factor 4) will result in a number greater than three-digits.

Edited on May 13, 2006, 8:04 pm

Posted by Dej Mar on 2006-05-13 19:51:35