Define the number representation

Any number $N$ can be expressed in decimal notation as $d_k 10^k + d_{k-1} 10^{k-1} +... + d_1 10^1 + d_0 10^0$.

$N = \sum_{i=0}^{n} d_i 10^i$

Apply modulo 9 property

Since $10 \equiv 1 \pmod{9}$, it follows that $10^i \equiv 1^i \equiv 1 \pmod{9}$ for any power $i$.

$N \equiv \sum_{i=0}^{n} d_i (1) \equiv \sum_{i=0}^{n} d_i \pmod{9}$

Conclusion based on remainder

If the sum of the digits $\sum d_i$ is divisible by 9, the remainder of $N$ when divided by 9 is 0, confirming divisibility.

$N \equiv 0 \pmod{9} \iff \sum d_i \equiv 0 \pmod{9}$