Rules for Divisibility

Proposition 2.2   A number $n\in\mathbb{Z}$ is divisible by $3$ if and only if the sum of the digits of $n$ is divisible by $3$.

Proof. Write

$$n=a+10b+100c+\cdots.$$

Since $10\equiv 1\pmod{3}$,

$$n = a + 10b + 100c+\cdots \equiv a + b + c+\cdots \pmod{3},$$

from which the proposition follows. $\qedsymbol$

Similarly, you can find rules for divisibility by $5$, $9$ and $11$. What about divisibility by $7$?