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?

I started with #4, which at first I believe to be true, which ends up being false.
But it lead me to come up with some numbers:  I know it will have to be a low number as adding 99 will reverse it, and that each digit is unique. 
Well if we add 99 and it reverses the number I figure the first number must be lower than the third number and they can't be more than 1 off of each other(as we are only adding basically 100).  If we want to switch the first and last(hence reversing the sequence) by adding, the first digit must be less than the second.
142 263  384   
1X2 2X3  3X4 where X is atleast the sum of the 2 outers +1.
All of these also have the property of adding 99 will reverse them. 
I was going through all of the rest of them in my head and thought this is too complicated.  So I thought some more and re-read the some are false. 
And I kept thinking, then I thought the number would be low, what if the first digit is 0. 
021 + 99 = 120 Bingo.
1) 2+1 = 3, 21 is divisible by 3
2) Not prime:  21 is divible so it most not be prime
3) one Common Digit with the product: 2*1*0 = 0, 1 zero
4) is false.  There is one false statement.
So the answer is "021"

Posted by John on 2006-04-22 00:48:05