To divide by something just use the sharing reasoning.
10 sweets divided by 2 people = 5 sweets each.
Because you can share 10 sweets between the 2 people.
10 sweets divided by 4 people = 2.5 sweets each
Because you can share 10 sweets between the 4 people, even if you don't get a whole number.
SO on an so forth, until you get to dividing by 0.
10 / 0 ... (sticking with the analogy of sweets)
You cannot share the sweets between noone, because there's noone to give the sweets to. Therefore dividing by 0 is not possible.
Using the example you tried:
0/0 = n
Yes, 0 = n x 0
but that would only work for the 0/0.
But what about if it was , say 7 / 0 = ?
then you would get 7 = ? x 0 Which wouldn't work.
You may then think that you have provided a counter example by saying 0/0 works.
However, 0/0 doesn't make sense. Why would you try and share (for example) sweets? There are no sweets to share, let alone people to share it between. WHich is why this is not a counter example as it is also an indeterminate..