Divisor of Integer/Examples/3 divides n(n+1)(n+2)

Theorem
Let $n$ be an integer.

Then:
 * $3 \divides n \paren {n + 1} \paren {n + 2}$

where $\divides$ indicates divisibility.

Proof
$n$ is of one of these forms:

for some $k \in \Z$.

Suppose $n = 3 k$.

Then $3 \divides n$ by definition.

Suppose $n = 3 k + 1$.

Then:
 * $n + 2 = 3 k + 3 = 3 \paren {k + 1}$

Thus:
 * $3 \divides n + 2$.

Suppose $n = 3 k + 2$.

Then:
 * $n + 1 = 3 k + 3 = 3 \paren {k + 1}$

Thus:
 * $3 \divides n + 1$.

Hence from Divisor Divides Multiple:
 * $3 \divides n \paren {n + 1} \paren {n + 2}$