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This may be a simple question and I am not a mathematician. I recently noticed that 11+12+13 = 36 whose digit sum is 9. And that the digit sum 1+1+1+2+1+3 also equals 9. Is there a pattern here that can be extrapolated to other addends?

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    $\begingroup$ The iterated digit sum (digital root, methinks it is called) gives you the remainder of the number modulo $9$. So the iterated digit sum of a sum always equals the iterated digit sum of the sum of digit sums of the addends. (Phew, a mouthful) $\endgroup$ – Daniel Fischer Nov 8 '13 at 18:59
  • $\begingroup$ @Daniel Good candidate for a tongue-twister. $\endgroup$ – Namaste Nov 8 '13 at 19:00
  • $\begingroup$ I remember noticing this pattern on license plates as a child and to this day I always test it when I see license plates (though I know it is true). In middle school I asked my math teacher about it and he gave me the same result that you are adding modulo 9. I think this pattern is very interesting though! $\endgroup$ – Jeremy Upsal Nov 8 '13 at 19:19
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The digit sums may differ: $99+2=101$ has digit sum $1+0+1=2$, but on the left we have $9+9+2=20$.

However, there is a pattern. The digit sum of a number differs from the number itself by a multiple of $9$ (that is why (repeated) digit sums can be used to check divisibility by $9$). Therefore the digit sum of the sum also differs by a multiple of $9$ form the digit sum of the summands. (So above: The difference between $20$ and $2$ is $18=2\cdot 9$)

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