Divisibility by 37

Theorem
Let $n$ be an integer which has at least $3$ digits when expressed in decimal notation.

Let the digits of $n$ be divided into groups of $3$, counting from the right, and those groups added.

Then the result is equal to a multiple of $37$ $n$ is divisible by $37$.

Proof
Write $n = \ds \sum_{i \mathop = 0}^k a_i 10^{3 i}$, where $0 \le a_i < 1000$.

This divides the digits of $n$ into groups of $3$.

Then the statement is equivalent to:
 * $37 \divides n \iff 37 \divides \ds \sum_{i \mathop = 0}^k a_i$

Note that $1000 = 37 \times 27 + 1 \equiv 1 \pmod {37}$.

Hence:

which proves our statement.