Exercise 1.3.4 (Divisibility criterion by 3,9)

Proof.

We use the same notations than in Problem 2, where $n=\overline{a_m{a}_{m-1}\cdots a_0}={\sum <!--\; FUNCTION\; APPLICATION\; -->}_{i=0}^m{a}_{i}10^i$.

Since $10\equiv 1(\mathit{mod}3)$, $n\equiv {\sum <!--\; FUNCTION\; APPLICATION\; -->}_{i=0}^m{a}_i(mod3)$, so $n$ is divisible by $3$ if and only if the sum of its digits is divisible by $3$.

Since $n\equiv 1(\mathit{mod}9)$, we have the same result with $9$. □