Each of p and q is a 6-digit base ten positive integer with no leading zero. The 12-digit number that is obtained by writing p and q side-by-side is divisible by the product p*q.

Determine all possible pair(s) (p, q) for which this is possible.

For one solution, there is a pattern.

Where p and q are both n+2 digits in length, and such that n > 0,
p, being a number of the form: 1x7, where x represents the digit 6 occuring n times, with q, being a number of the form: 3y4, where y represents the digit 3 occuring n times; the product,  p*q, would be of the form: wz8, where w represents the digit 5 occuring n+1 times and z represents the digit 7 occuring n+1 times. For the numbers of this pattern, p|q / p*q = 3.
Thus, (166667, 333334) is one solution where the 12-digit number 166667333334 / 55555777778 = 3.

I had submitted this as a partial solution as I had not yet performed a thorough check for other solutions. After reading Steve Herman's post, I was able to complete this check in a relative short time. The solution, (166667, 333334), is unique.  

Edited on June 29, 2010, 4:41 pm

Posted by Dej Mar on 2010-06-29 16:06:57