Question

Show that any number of the form 4 raise to N belong to natural number can never end with a digit zero

Answer 1

4^n = 2^n * 2^n thus the only prime in the factorisation of 4^n is 2 and 2 and the uniqueness of fundamental theorum of arithmatic guarentees 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 contain primes 5 and 2 . but this is not possible as mentioned above so there is no natural number n for whch 4^n ends with digit 0 [ to understand this question in much better way try to understand fundamental theorum ]