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 \(n?\)