Question
Check whether $6^n$ can end with the digit $0$ for any natural number $n$.
✓
Answer
If the number $6^n$ ends with the digit zero, then it is divisibli by $5$.
Therefore the prime factorization of $6^n$ contains the prime $5$.
This is not possible because the only prime in the factorisation of $6^n$ is $2$ and $3$ and the uniqueness of the fundamental theorem of arithmetic guarantees that these are no other prime in the factorisation of $6^n$.
Hence it is very clear that there is no value of $n$ in natural numbers for which $6^n$ ends with the digit zero.
Therefore the prime factorization of $6^n$ contains the prime $5$.
This is not possible because the only prime in the factorisation of $6^n$ is $2$ and $3$ and the uniqueness of the fundamental theorem of arithmetic guarantees that these are no other prime in the factorisation of $6^n$.
Hence it is very clear that there is no value of $n$ in natural numbers for which $6^n$ ends with the digit zero.
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