Products, sums and divisors

66 is divisible by 66.
6×66\times6 is divisible by 6+66+6.
6×6×66\times6\times6 is divisible by 6+6+66+6+6.

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

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