Question

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

Answer 1

If a number ends with the digit zero then it will be divisible by 2 and 5 both. So here first write the given number as the product of prime factors and then check this prime factorization contains 2 and 5 or not.If it contains 2 and 5 both, then given number ends with the digit 0, otherwise not.

SOLUTION:

Given, n is a natural number.

Let 6ⁿ ends with 0, then 6ⁿ is divisible by 2 and 5. But prime factors of 6 are 2×3

6ⁿ = (2×3)ⁿ  

Thus, prime factorization of 6ⁿ does not contain  5. so the uniqueness of the fundamental theorem of arithmetic guarantees that there is no other primes in the factorization of 6ⁿ.  

Hence 6ⁿ can never ends with digit 0 for any natural number.

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