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Equal digit sum (Posted on 2012-11-22) Difficulty: 5 of 5
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    
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Solution re: The first answer (spoiler) Comment 4 of 4 |
(In reply to The first answer (spoiler) by Steve Herman)

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

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