Question

For any natural number $n,4^n$ cannot end with which of the following?

• A. 4
• B. 6
• C. \N
• D. None of the above

Answer 1

The correct option is C \N

For $n=1,$
$4^n=4^1=4.$
For $n=2,$
$4^n=4^2=16.$
For the number $4^n$ to end with digit zero for any natural number $n,$ it should be divisible by both 2 and 5. This means that the prime factorisation of $4^n$ should contain the prime numbers 2 and 5. But, this is not possible because $4=2^2$ so 2 is the only prime in the factorisation of $4^n.$ Since 5 is not present in the prime factorization, so there is no natural number $n$ for which $4^n$ ends with the digit zero.