```On Fri, 02 May 2008 13:18:37 -0500, Martin DeMello wrote:

> On Fri, May 2, 2008 at 10:25 AM, Matthew Moss <matthew.moss / gmail.com>
> wrote:
>> > 1089 * 9 = 9801
>>  > 2178 * 4 = 8712
>>  > 10989 * 9 = 98901
>>  > 21978 * 4 = 87912
>>  > 109989 * 9 = 989901
>>  > 219978 * 4 = 879912
>>
>>  One of the interesting things I found is that those are all divisible
>>  by 9. I wonder if that is a property of all such numbers?
>
> Let the numbers be x = a....c and y = c....a
>
> then, if y divides x, so does (x-y)
>
> x = 10^n a + 10b + c
> y = 10^n c + 10 d + a
>
> x - y = (a - c)(10^n-1) + 10 (b-d)
>
> the first term obviously divides by 9
>
> for the second term, note that b and d are reversals of each other. It
> can be shown that their difference again divides by 9 (again splitting
> off the first and last digit as above, or simply by induction on the
> number of digits, which come to think of it I should have done from the
> beginning)
>
> martin

I think this may fall into the category of a spoiler.

--Ken

--
Ken (Chanoch) Bloom. PhD candidate. Linguistic Cognition Laboratory.
Department of Computer Science. Illinois Institute of Technology.
http://www.iit.edu/~kbloom1/

```