N is a positive integer and S(N) denotes the sum of the digits.

Find all values of a positive integer constant C such that:
S(N) = C*S(N+3)

Unless I'm overlooking something: C can be any number that is not one more than a multiple of 9.

How to construct a number with a given C:
Suppose you want C=22.
22 is one more than 21.  A multiple of 21 is of the form 9a-3, specifically 21*2 = 9*5-3.  a=5 is the number of carries.   2 is a number I'll call the shortfall:  the sum of the difference between each digit and 9.
Then N=99998 and N+3=100001 so that 44=22*2.

Multiples of 9 cannot be written as 9a-3 so C cannot be one more than a multiple of 9.

Apparently I need to take my son to a swimming hole so I'll need to flesh this out when I get a chance.

Posted by Jer on 2016-05-28 14:24:16