I brought this one up in HS and every honors math teacher had a different approach.

Algebraically, if we set it up as x=0/0 then 0=0x, which is true for any and all values of x.

If we use calculus to replace 0 with a variable x that approaches 0 we can put it two different places, which kinda makes things worse. Lim x/x as x→0 is 1; until/unless x actually REACHES 0 it's always 1. Lim 0/x as x→0 is 0; again, until/unless x reaches 0 it always will be. Lim x/0 as x→0 is our old friend Mr. Undefined; until/unless x reaches 0 it can't be anything else and, as is hopefully becoming clear, things don't really improve then. If you want to get fancy and use the lim 1/x as x→∞ for 0 we get lim (1/x)/ (1/x) as x→∞, or 0/0=lim x/x as x→∞=1.

Logically, how many times can nothing be divided among nothing? Depending perspective, as many as you like, because there's no place to divide it, none, because there's nothing to divide, or exactly one, by definition.

0/0 is undefined because it has too many solutions, all in the eye of the beholder. I favor an algebraic answer that says 0/0=the set of all numbers, but that's ultimately no more or less right than any other. It's one of the uncommon but significant cases revealing the fallacy that math can conveniently and precisely define everything, which is what makes it an interesting topic. Speaking of interesting topics yielding the set of all numbers, I find 3^-4 entertaining also.