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

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

We are asked to check whether $6^n$ can end with the digit 0 for any natural number $n$.

For a number to end with the digit 0, it must be divisible by 10. We know that:
$10=2\times 5$

Thus, for $6^n$ to end in 0, it must be divisible by both 2 and 5.

Now, observe the prime factorization of 6:
$6=2\times 3$

Hence, $6^n=(2\times 3{)}^n=2^n\times 3^n$.

For $6^n$ to be divisible by 5, it must have 5 as one of its prime factors. However, the prime factorization of $6^n$ contains only the primes 2 and 3, but not 5. Therefore, $6^n$ can never be divisible by 5, and thus it cannot end in 0.

Conclusion:
$6^n$ can never end with the digit 0 for any natural number $n$.

Thanks for awesome explaination.

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