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Question

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

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Solution

If the number 6n, for any natural number n, ends with digit 0, then it would be divisible by 5. That is, the prime factorization of 6n would contain the prime number 5. This is not possible because 6n=(2×3)n=2n×3n ; so the only primes in the factorization of 6n are 2 and 3 and the uniqueness of the Fundamental Theorem of Arithmetic guarantees that there are no other primes in the factorization of 6n. So, there is no natural number n for which 6n ends with the digit zero.


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