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

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

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Solution :

If the number 6n ends with the digit zero,then it is divisible by 5.Therefore the prime factorization of 6contains the prime 5.This is not possible because the only prime in the factorization of 6n is 2 and 3 and the uniqueness of the fundamental theorem of arithmetic guarantees that these are no other prime in the factorization of 6n

Hence, it is very clear that there is no value of n in natural number for which 6n ends with the digit zero.

by (128k points)

If any number ends with the digit 0, it should be divisible by 10 or in other words, it will also be divisible by 2 and 5 as 10 = 2 × 5
Prime factorisation of 6n = (2 ×3)n
It can be observed that 5 is not in the prime factorisation of 6n.
Hence, for any value of n, 6n will not be divisible by 5.
Therefore, 6n cannot end with the digit 0 for any natural number n.