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 nonzero distinct digits.
Find the two existing solutions to this 9digit 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*ei=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 20100915 23:27:00 