Question

Show that any no. of the form $4^n$ can never end with 0.

Answer 1

$4^n=2^n\times 2^n$ thus the only prime in the factorisation of $4^n$ is 2 and 2 and the uniqueness of fundamental theorem of arithmetic guarantees that there are no other primes in factorisation of $4^n$ except 2 and 2.
If the number $4^n$ for any n, ends with digit zero then it has to be divisible by 5 and 2 both that is prime factorisation of $4^n$ has to conatin primes 5 and 2.
But this is not possible as mentioned above
So there is no natural number n for which $4^n$ ends with digit 0.