Products, sums and divisors

\(6\) is divisible by \(6\).
\(6\times6\) is divisible by \(6+6\).
\(6\times6\times6\) is divisible by \(6+6+6\).

Is it true that \(\underbrace{6\times6\times\cdots\times6}_{n\text{ times}} \) is divisible by \(\underbrace{6+6+\cdots+6}_{n\text{ times}} \) for all positive integers \(n\)?

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