From: Shane Emmons Date: 2008-05-03T03:56:10+09:00 Subject: Re: Reverse Divisible Numbers (#161) ------=_Part_12330_31953538.1209754563680 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Content-Disposition: inline On Fri, May 2, 2008 at 2:18 PM, Martin DeMello wrote: > On Fri, May 2, 2008 at 10:25 AM, Matthew Moss > 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 > > With that info, I was able to get the code to run under 1 sec. === c:\ruby-1.9.0\bin\ruby.exe quiz161.rb === 8712 9801 87912 98901 879912 989901 Execution time: 0.969 s -- Shane Emmons E: semmons99@gmail.com ------=_Part_12330_31953538.1209754563680--