Posted: Sun May 09, 2004 3:53 pm    Post subject: “sum of digits” functional equation

Let f(n) denote the sum of (all) digits of natural number n, e.g. f(2004)=2+0+0+4=6. Prove that for each natural n we can choose convenient value of natural parameter p such that the equation f(npx)=f(x) has solution in natural numbers x that doesn’t contain any “9” in its notation (of course in base 10). Does anybody have any idea? I don’t… but I hope that you do… Actually I can solve a lot of special cases on a lot of pages… but I can’t solve it generally. Is there any trick or only hard work?

Thank you very much…