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# Congruent modulo numbers watch

1. Just a little sentence that I can't quite grasp:

Because 9 is the largest digit in the decimal number system, the sum of the digits of any number will always be congruent modulo 9 to the original number.

Could someone explain this a little more fully please?
2. How familiar are you with modulo arithmetic?
3. (Original post by Erdős)
The statement is strangely explained (I believe).

Is it more obvious if one says: ?
Well that's clearly true, but it doesn't explain why the sum of the digits of any number has the same mod as the number itself?

I'm not too familiar with modulus arithmetic, just reading about it in little bits here and there..
4. How about we write our number in two ways - the first as a sum of digits, the second as each digit multiplied by its units. Using the facts that:

and:

We can prove equality here.

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