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### Proving that $9$ is a divisor of $x \in \Bbb N$ if the sum of digits of $x$ is divisible by $9$. [duplicate]

Suppose x is a positive integer with $n$ digits, say $x = d_1d_2d_3\ldots d_n.$ If $9$ is a divisor of $d_1 + d_2 + \ldots d_n$, prove then $9$ is a divisor of $x$. My attempt: suppose $x = 4518.$ ...
Im trying to prove that every natural number is divisble by three if and only if the sum of its digits are divisible by three. First i proved by induction that $10^n-1$ is divisible by 9 (and ...