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.
I decided to approach this in small steps to see if I could find a pattern which could be extended.
If the two numbers were each 2-digits, we have two solutions:
13 and 52 (1352 divisible by 676) and
17 and 34 (1734 divisible by 578).
If the two numbes were each 3-digits, we have two solutions if the numbers can be the same:
143 and 143 (143143 divisible by 20449)
167 and 334 (167334 divisible by 55778).
I notice that in all these cases the righthand number is a product of the lefthand. If there is a good ("analytic") reason why this must be the case (e.g. limits on factorization options), the brute force approach could be reduced (a full search for even two 4-digits is extensive). Bring in the Arithmetica crowd!
prelims
