Showing that $3^{(3^n - 1)}$ divides $(3^n)!$

Question

I am trying to solve the following by using induction:

Show that $3^{(3^n - 1)}$ divides $(3^n)!$ for any non-negative integer $n$.

But isn't the question incorrect, since it doesn't hold for $n=1$, or am I missing something?

Answer 1

I am afraid that the problem is incorrect.

The highest power of 3 that devides $(3^n)!$ is $\sum _{ i=1 }^{ \infty }{ \left\lfloor \frac { 3^{ n } }{ 3^{ i } } \right\rfloor } $, which is equal to $\frac { 3^{ n }-1 }{ 2 } $.

The proof to the general highest power of $p$ when p is a prime number that divides $n$! can be seen here.