Find a four-digit number with four different digits, that is equal to the number formed by its digits in descending order minus the number formed by its digits in ascending order.

(In reply to divisible by 9 by abc)

I was going to submit a question about "prove the divisibility rules", but we have queue problems already, so I won't ;)

The proof for any number like this that will loop with itself (like 6174) is any number minus a number with the same digits is divisible by 9.

Starting with the lesser number| 1467


Add a 9 on each time until the | 1476


one's place is the digit you   | 1485


want, which is the highest     | 1494


number that uses those digits  | 1503


Note that the numbers go up or | 1512


down when we subtract 9s       | 1521





To the tens place. Subtracting | 1521


on 9x10 is the same idea, only | 1431


modified for the ten's place   | 1341





For the hundred's place we need| 1341


to add on 900s so we get the   | 2241


number on the hundred's place  | 3141


that we want. Since we can't   | 4041


subtract 900s (the quicker way)| 4941


because we are going up instead| 5841


of going down, we must use more| 6741


numbers and add on more 900s.  | 7641

We end with 7641, the number we wanted.
Since we added on 9s each time using this process, the difference between the two numbers must be a multiple of 9. Since this process can be used for any number of 9s, this is a proof for all of them.

Posted by Gamer on 2003-09-12 17:36:20