From: Martin DeMello <martindemello@...>
Date: 2008-05-03T03:18:37+09:00
Subject: Re: Reverse Divisible Numbers (#161)

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