Divisibility by 9/Proof 2

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Theorem

A number expressed in decimal notation is divisible by $9$ if and only if the sum of its digits is divisible by $9$.


That is:

$N = \sqbrk {a_0 a_1 a_2 \ldots a_n}_{10} = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$ is divisible by $9$

if and only if:

$a_0 + a_1 + \ldots + a_n$ is divisible by $9$.


Proof

This is a special case of Congruence of Sum of Digits to Base Less 1.

$\blacksquare$