Question

Check whether $4^n$ can end with 0 or not.

Answer 1

If the number $4^n$, for any $n$, were to end with the digit zero, then it should be divisible by 5. That is, prime factorisation of $4^n$ would contain the prime factor 5. This is not possible because $4^n=(2^2{)}^n=2^{2n}.$ So, the only prime in the factorisation of $4^n$ is 2.

So, by uniqueness of the fundamental theorem of Arithmetic, there are no other primes in the factorisation of $4^n$.

Hence, there is no natural number 'n' for which $4^n$. ends with the digit zero.