No.You wont be able to completely divide m by q, because remember

hence, you will always get the remainder 1.If,say,the random ,and we divide (without ) or 1 is the remainder as it does not fully go with .

It comes because of the above reason.

You have already supposed that are your only primes, and you have no more of them.But you have actually built yourself a new prime number , because it has all the properties of prime(isn't divisible by anything other than 1 or itself), and it ought to be on your list.So you have one more prime..and the same reason could be applied to get more and more of such primes...to infinity.

I suggest to take and work with some examples like make your list of primes only and see what happens....