Products, sums and divisors

Number Theory Level 1

$6$ 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 $n$ ?