There is a four digit number such that when multiplied by a single digit produces another four digit number i.e. "ABCD"*E="FGHI " (the letters between the quotation signs designate a concatenation, not the product).

The letters represent 9 non-zero distinct digits.

Find the two existing solutions to this 9-digit problem.

We know from the fact that the multiplication caused ABCD to change to another number, that e is not 1, nor can it be 5, because I would then be 5. It is also obvious that d cannot be 1 or 5. Now d*e-i=10 k1 from which it is readily apparent that e, d and i are all odd or i and at least one of (d,e) are even. We can also exclude the solutions where e = 9, the reason being that even if a and b were 1 and 2, there is still an overflow to 5 figures when 12 is multiplied by 9. i cannot be 1 if e is larger than 4, since then a would also need to be 1. The remaining odd possibilities (e,d,i,k1) where k1 is the first carry are (3,7,1,2)(3,9,7,2) and (7,9,3,6); these can easily be excluded by checking.

Applying the same reasoning, the candidates (e,d,i,k1) where e,i are even are: (2,3,6,0)(2,4,8,0)(2,7,4,1)(2,8,6,1)(2,9,8,1)(4,2,8,0)(4,3,2,1)(4,7,8,2)(4,8,2,3) (4,9,6,3)(6, 3, 8,1)(6,7,2,4)(8,2,6,1)(8,3,4,2)(8,4,2,3)(8,7,6,5).Checking these gives the two solutions :

 A B C D 
 1 7 3 8 
         4 E
 2 1 3   
 6 9 5 2 
 F G H I 
     
 A B C D 
 1 9 6 3 
         4 E
 3 2 1   
 7 8 5 2 
 F G H I 
     

Edited on September 17, 2010, 12:36 am

Posted by broll on 2010-09-15 23:27:00