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the sum of the digits watch

1. Let f(x) be the sum of the digits of x. Find.
2. Hmm, is f(19) = 10 or f(19) = 1? If it's the latter, there's a trick to simplify the calculation significantly.
3. (Original post by Zhen Lin)
Hmm, is f(19) = 10 or f(19) = 1? If it's the latter, there's a trick to simplify the calculation significantly.
The former, I'm pretty sure...
4. Spoiler:
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This is a fairly standard question, actually, and Zhen Lin is on the right lines.

Step 1:

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As you probably know, f(x) = x (mod 9). Let N = 8765^4321. The first step is to find f(N) mod 9. I assume you know how to do this.

Step 2:

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The next step is to put a bound on f(f(f(N))). We want to show f(f(f(N))) is pretty small, so that knowing f(f(f(N))) mod 9 tells us what f(f(f(N))) is.

Step 2a:

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How many digits are there in N? So what's an upper bound for f(N)?

Step 2b:

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So what's an upper bound for f(f(N))? And a bound for f(f(f(N)))?

Step 3:

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Combining steps 1 and 2, what possible values are there for f(f(f(N)))?

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Updated: March 11, 2008
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