OK seriously…
Yes anything times zero is zero. And if you have pq=0 then you know either p is 0 or q is 0 or both.
Another way to think about multiplication is that you are making combinations. If I had 3 colors and 4 shapes then I have 3×4 = 12 combinations of colored shapes.
But if I had 5 shapes and no colors I would have zero combinations of ‘colored shapes’ since…there are no colors.
And that’s the nice thing about multiplication, because you can take say velocity and time, multiply them together, cancel the units and get the distance.