An odd coincidence?

Algebra Level 1

$1$ is divisible by $1$ .
$1\times2\times3$ is divisible by $1+2+3$ .
$1\times2\times3\times4\times5$ is divisible by $1+2+3+4+5$ .
$1\times2\times3\times4\times5\times6\times7$ is divisible by $1+2+3+4+5+6+7$ .

In general, is it true that $1\times2\times\cdots\times n$ is divisible by $1+2+\cdots+n$ for all odd $n?$