The fact that you're calling these "delta functions" in the title seems to indicate that you're dealing with these things in a setting where rigour isn't emphasized. In physics, delta distributions are often treated like ordinary functions, more often than not with sensible results.
However, your question seems to be looking for more rigour. A rigorous treatment of what you call "delta functions" leads to distribution theory. Distributions can be regarded as generalizations of ordinary functions, but they don't share all the properties of ordinary functions.
In particular, as user8268 pointed out in the comments, multiplication of distributions works differently. The product of two general distributions (e.g. the product of two delta distributions) cannot be defined (at least not in a straightforward sense; see here). The product of a distribution with an ordinary function can be defined, but as user8268 pointed out, this is not invertible. Division is the inverse of multiplication, so if multiplication is not invertible, division makes no sense. If you applied your professor's division rule to user8268's example, you could cancel the delta distributions and get $1=1+x-a$, which is obviously wrong.
I'd suggest that you look at what your professor was trying to use this relationship for and see whether it can be justified rigorously without resorting to division of distributions. For instance, if you have two charge distributions and you know that they're both localized on a sphere of radius $R$, you can write them as $\rho_i(\theta,\phi)\delta(r-R)$; then if by some argument you find that they're the same, $\rho_1(\theta,\phi)\delta(r-R)=\rho_2(\theta,\phi)\delta(r-R)$, you can conclude that $\rho_1(\theta,\phi)=\rho_2(\theta,\phi)$ by other means than "dividing through" by the delta distribution.