Let
![a,b,n](/media/m/a/4/c/a4c62fb9a19029f05a846bbde965ddea.png)
be positive integers,
![b > 1](/media/m/8/5/b/85b6cfab63ac4b9adf62b72ffc296082.png)
and
![b^n-1|a.](/media/m/0/e/c/0ec097f4de12f6d1311e625073feb254.png)
Show that the representation of the number
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
in the base
![b](/media/m/e/e/c/eec0d7323095a1f2101fc1a74d069df6.png)
contains at least
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
digits different from zero.
%V0
Let $a,b,n$ be positive integers, $b > 1$ and $b^n-1|a.$ Show that the representation of the number $a$ in the base $b$ contains at least $n$ digits different from zero.