All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars
 perplexus dot info

 Equal digit sum (Posted on 2012-11-22)
Find all integers n such that S(n) = S(2n) = ... = S(n*n), where S is the sum of the base-10 digits.

 No Solution Yet Submitted by Danish Ahmed Khan No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
 re: The first answer (spoiler) Comment 4 of 4 |

You say

////Arguably 0 and 1 work also, but I think the form of the problem implies that N >=2.
/////
0  -yes  1 - no

The answer is 9, 99, 999 ,,, 10^k-1      k=0,1,2,3...
The general proof  is tiresome  and it involves to many indices , so I will just show why it works for 99 multiples
0f 99:
(10a+b)(99)=(10a+b)(100-1)=1000a+100(b-1)+10(9-a)+10-b
the sod of the right side is a+b-1+9-a+10-b=-1+9+10=18 like sod 0f 99.

e.g. 99*64=6400-64=6336,  6+3+3+6=18

Edited on November 22, 2012, 4:46 pm
 Posted by Ady TZIDON on 2012-11-22 16:26:53

 Search: Search body:
Forums (0)