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Funny Factorisation

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Some 4 digit numbers can be written as the product of a 3 digit number and a 2 digit number using the digits 1 to 9 each once and only once.

Well done Sally Nelson and Sarah Dunn, S2, Madras College, St Andrew's for finding altogether six funny factorisations, but there is one more. It is now a Tough Nut to find the last one. You might like to write a computer program to find all seven funny factorisations or you might come up with a different method. Let us know.

The number 4396 = 2 x 2 x 7 x 157 and there are not many possible combinations. By trial and error we get 4396 = 28 x 157.

The number 5796 = 2 x 2 x 3 x 3 x 7 x 23.
So 5796 = (2 x 3 x 7) x ( 2 x 3 x 23) or (2 x 2 x 3) x (3 x 7 x 23) amongst other possibilities which don't turn out to be 'funny'.
In this way we find the two funny factorisations: 5796 = 42 x 138 and 5796 = 12 x 483.

Similarly 5346 = 2 x 3 5 x 11 and the funny factorisations are:
5346 = 27 x 198 and 5346 =18 x 297.
 

Here you must use the digits 1 to 9 once, but only once, to replace the stars and complete this multiplication example.

  * * 9
$\times$   4 *
--- --- --- ---
* 6 * *

Firstly I found out the possible solutions for the top row. It could not be a number above 250 or below 100 and it had to end in a 9. The number could not have a 4 or a 6 or another 9. The only possibilities were 129, 139, 159, 179, 189, 219 and 239. So I tried these numbers with every 2 digit number beginning with a 4 until I found the answer 159 x 48 = 7632.


We received a Python program from Ryan for exhaustively finding solutions to the problem. You can download it here