Check whether $6^n$ can end with the digit 0 for any natural number n.

Given: $6^n$.

To do: Here we have to check whether $6^n$ can end with the digit 0 for any natural number $n$.

Solution:

There is no natural number  $n$  for which  $6^n$  ends with the digit zero.

Explanation:

If  $6^n$  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  $6^n$  should contain the prime number 5 and 2.

But it is not possible because;

$6^n\ =\ (2\ \times\ 3)^n\ =\ 2^n\ \times\ 3^n$

Since 5 is not present in the prime factorization, there is no natural number  $n$  for which  $6^n$  ends with the digit zero.

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Updated on: 10-Oct-2022

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