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 $9$
- $a_0 + a_1 + \cdots + a_n$ is divisible by $9$.
Proof
This is a special case of Congruence of Sum of Digits to Base Less 1.
$\blacksquare$