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.
OK, yes, I made a mistake in my analysis. Here is a revised solution:
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.
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)/kn
Since p is integral, k must divide 1000000.
k must be 1,2,4,5 or 8
5) This leaves a small number of possibilities, when k divides 1000000 and n divides (1000000 + k):
k 1000000 + k n q p
-- -------------- --- --------- ---------
2 1000002 3 333334 166667 <-- only solution
2 1000002 6 166667 83333.5 <-- p is not integral
4 1000004 4 250001 62500.25 <-- p is not integral
5 1000005 5 200001 40000.2 <-- p is not integral
8 1000008 3 333336 41667 <-- p too small
8 1000008 4 250002 31250.25 <-- p is not integral
8 1000008 6 166668 20833.5 <-- p is not integral
8 1000008 8 125001 15625.125 <-- p is not integral
8 1000008 9 111112 13889 <-- p too small
So, Dej Mar's solution is the only one
Full solution
