An odd coincidence?

Algebra Level 1

11 is divisible by 11.
1×2×31\times2\times3 is divisible by 1+2+31+2+3.
1×2×3×4×51\times2\times3\times4\times5 is divisible by 1+2+3+4+51+2+3+4+5.
1×2×3×4×5×6×71\times2\times3\times4\times5\times6\times7 is divisible by 1+2+3+4+5+6+71+2+3+4+5+6+7.

In general, is it true that 1×2××n1\times2\times\cdots\times n is divisible by 1+2++n1+2+\cdots+n for all odd n?n?

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