If the number 4^{n} for any n, were to end with the digit Zero then it would be divisible by 5 and 2.that is prime factorization of 4^{n} would contain the prime 5 and 2. This is not possible beacuse 4n =(2)^{2n}; so the only prime in the factorization of 4^{n} is 2. So, the uniqueness of the Fundamental Theorem of Arthmetic guarantees that there are no other primes in the factorization of 4^{n }. So, there is no natural number n for which 4^{n} ends with the digit zero.