A boy met his father after his maths exam. He said, "Dad can you guess the number of students who appeared at our center?"

The following are true regarding the number
a) It has three distinct digits.
b) If you add 99 the number reverses.

He continues saying, "I have the following statements to add regarding the number."

1) It is divisible by the sum of its digits.
2) It is not prime.
3) It has only one common digit with the product of the digits.
4) The sum of the first and last digit is one more than the middle digit.

He adds further that "if I told you which of these statement(s) is/are false then you'd be able to determine the number."

Dad got it. What was the number?

OK, so we're trying to find a three digit number, and when you add 99 the number reverses.  This tells us that the 3rd digit is one more than the first digit.  That narrows it down a bit.

Of the second set of statements, only one of them seemed like it would narrow down the field to a manageable size - clue #4.  If we assume that this is true, then there are only three possible combinations: 243, 364, and 485.

Of these three, none are prime, so clue #2 isn't helpful, and none of them has only one common digit with the product of its digits, so clue #3 is equally useless.  However, 485 is the only one that is not divisible by the sum of its digits, so this must be the number we are looking for.

Bear in mind that I only assumed clue #4 was true.  I suppose it might be possible to find a different unique solution if it is false - however, I personally am pretty confident that it isn't possible, so I'm not going to look for it. :)

Posted by tomarken on 2006-04-21 12:59:47