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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 $n$.
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