I was taught in school that it is forbidden to divide by zero. Every time I had to exclude this option in mathematical equation reduction. I was just following the rules.
But at some point I've earned enough experience with people making a very special kind of error in almost every possible field. They were failing to identify that some operations they consider as atomic, in fact consist of several inner operations, which applied implicitly one after another. There are things impossible to understand before you learn to identify such sequences.
Lets identify inner operations in dividing.
The operation of dividing X by Y is defined as an answer to a question: "What number multiplied by Y gives X?" In every case you see this structure in a question the answer is a group, not a number. For example, a square root of X is defined as an answer to a question: "What number multiplied by itself gives X?" So, square root of 4 is {2, -2}, which is a group of two numbers: 2 and -2. In this manner the result of dividing 6 by 2 would be a group of one number: {3}
But then, we usually perform additional numerical operations with the result, while having a group at hand. To solve this issue we've defined implicit conversion from a group of single number to that number, which works fine in most of the cases.
Now, what happens when you divide 5 by 0? The answer to the corresponding question "What number multiplied by 0 gives 5?" is an empty group: {}
Because there is no such number!
There is nothing wrong, impossible or difficult in dividing by 0. The problem arises when you try to perform further numerical operations with the result, since there is no defined conversion from an empty group to a number.
By now you should easily find the answer to the following question: "What is the result of dividing 0 by 0?"