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Given a natural number $x$, $f(x)$ adds all the digits of $x$ to make a new number, then adds all the digits of the new number, and continues to do so until we are left with a one-digit number.

For example, $f(139)=4$

Prove: $$ f(f(a)+f(x-a))=f(x) $$ When $a,x,(x-a)\in N$

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    Did you compute $f(x)$ for some integers? Can you observe a simple pattern? – 2017-01-01
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    Ah! $f(x)$ takes the values {1, 2, 3, 4, 5, 6, 7, 8, 9} over each interval of 9... – 2017-01-01
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    Hint: $f(x) = x \bmod 9$. See [this](http://www.cut-the-knot.org/Generalization/div11.shtml) for example. – 2017-01-01
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    The resulting $f(x)$ is known as the [digital root](https://en.wikipedia.org/wiki/Digital_root) of $x$. – 2017-01-01
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    Your question was put on hold, the message above (and possibly comments) should give an explanation why. (In particular, [this link](http://meta.math.stackexchange.com/a/9960) might be useful.) You might try to edit your question to address these issues. Note that the next edit puts your post in the review queue, where users can vote whether to reopen it or leave it closed. (Therefore it would be good to avoid minor edits and improve your question as much as possible with the next edit.) – 2017-01-02

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