Consider the numbers $ 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.

Given: 

$4^n$.

To do: 

Here, we have to check whether $4^n$ can end with the digit 0 for any natural number $n$.

Solution:

There is no natural number $n$ for which $4^n$ ends with the digit zero.

Explanation:

If $4^n$ is to end with zero for a natural number $n$, it should be divisible by 2 and 5.

This means that the prime factorisation of $4^n$ should contain the prime number 5 and 2.

But it is not possible because;

$4^n\ =\ (2\ \times\ 2)^n\ =\ 2^n\ \times\ 2^n$

Since 5 is not present in the prime factorization, there is no natural number $n$ for which $4^n$ ends with the digit zero.

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Updated on: 10-Oct-2022

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