Question

Prove that any number in form 6^n not end with zero

Answer 1

prime factorisation of 6^n= 2^n × 3^n
as we know that for a number to end with zero it must have factors only 2 and 5.as 5 is not the factor of 6^n and also 3 is included in its prime factorization. therefore we can say that 6^n cannot end with a digit zero.

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Answer 2

The number is of the form $6^{n}$, where $n\in\mathbb{N}$

Now, $6^{n}$

$\quad=(3\times 2)^{n}$

$\quad=3^{n}\times 2^{n}$

Here n > 0 and we also know that no exponential value of 2 or 3 ends with a 0.

Reason:

$\quad\quad 2^{n}=2,4,8,16,32,...$

$\quad\quad 3^{n}=3,9,27,81,243,...$

Therefore, $6^{n},\:n\in\mathbb{N}$ is not divisible by 10, and a number which is not divisible by 10, cannot end with a 0.

This completes the proof.

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