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