Question

Consider the number $4^n$, where n is a natural number. Check whether there is any value of n for which $4^n$ ends with the digit zero. [2 MARKS]

Answer 1

Concept : 1 Mark
Application : 1 Mark

If the number $4^n$, for any n, were to end with the digit zero, then it would be divisible by 5.

That is, the prime factorization of $4^n$ would contain the prime 5.

But, $4^n=(2\times 2{)}^n$

$\Rightarrow$ The only prime in the factorization of $4^n$ is 2.

So, the uniqueness of the Fundamental Theorem of Arithmetic 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.