Show that $12^n$ cannot end with the digit 0 or 5 for any natural number $n$.

To find: 

We have to show that $12^n$ cannot end with the digit 0 or 5 for any natural number $n$.

Solution:

We know that,

If a number ends with the digit 0 or 5, it is always divisible by 5.

This implies,

If $12^n$ ends with the digit zero it must be divisible by 5.

This is possible only if the prime factorisation of $12^n$ contains the prime number 5.

Prime factorisation of 12 is,

$12=2\times2\times3$

$\Rightarrow 12^n=(2\times2\times3)^n$

$=2^{2n}\times3^n$

The prime factorisation of $12^n$ does not contain the prime number 5.

Therefore, $12^n$ cannot end with the digit 0 or 5 for any natural number $n$.