NFA that accepts all numbers not divisible by 105
Here is a hint. 

$L$ is the union of the following three languages.


$L_3=\{{a^n \mid n\ge1\text{ and }n \not=0 \mod 3}\}$,
$L_5=\{{a^n \mid n\ge1\text{ and }n \not=0 \mod 5}\}$,
$L_7=\{{a^n \mid n\ge1\text{ and }n \not=0 \mod 7}\}$.


Construct the DFA for the above three languages. Then combine them together.



Exercise. (It may take some time to do this exercise.) Show that no NFA with less than 15 states accepts $L$ exactly.