If p = (3)(5)(6)(9)(q) and if q is a positive integer, then p must be

We can prime factorize on the right side of the equation:

p = (3)(5)(6)(9)(q) = (3)(5)(2)(3)(3^2)(q) = (2)(3^4)(5)q

We know nothing about q (for all we know, q = 1), so all we know for sure about p is that it is divisible by (2)(3^4)(5), and thus by any divisor of (2)(3^4)(5). We can see that 27 = 3^3, 45 = (3^2)(5), 54 = (2)(3^3) and 90 = (2)(3^2)(5) are all divisors of (2)(3^4)(5), but 36 = (2^2)(3^2) is not because we don't have enough 2's in (2)(3^4)(5), and so we can't be sure p is divisible by 36 (it will be if q is even, but will not be if q is odd).

If this is your question, Bunuel, you can erase the phrase "with no remainder" from the stem, because the phrase "divisible by" already means, definitionally, that there is no remainder.