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.
1) n = (1000000p + q)/pq , where n is a integer
2) Multiplying by q,
qn = 1000000 + (q/p)
Since qn is an integer, so is q/p.
(Yes, Ed, q must be a multiple of p)
Let q/p = k, where k must be an integer between 1 and 9.
3) Since q is between 100000 and 999999
And qn is between 1000001 and 1000009
It follows that n must be between 2 and 10
4) q = (1000000 + k)/n
p = (1000000 + k)*(k/n), so k must be less than n
This led to a relatively short search. Unless I made a mistake in Excel, there are no solutions.
More analytics
