Fundamental Theorem of Arithmetic (FTA) states that every composite number can be uniquely expressed as a product of prime numbers.
Using FTA, we can say that any number ending with $0$ must have both $2$ and $5$ as prime factors.
$\hspace{2.65cm}6=2\times 3$
$\hspace{2cm}(6)^{n}=(2\times 3)^n$
$\hspace{2cm}(6)^{n}=(2)^n\times (3)^n$
Using FTA, we can say that $6^n$ does not have $5$ as a prime factor.
So, $6^n$ cannot end with the digit $0$ for any natural number $n$.