Answer:

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.

Step-by-step explanation:

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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.