[Satan]

It's a pretty nice system for something imaginary.

Whichever way you look at it, the algorithm that's used to divide, doesn't work when something is divided by 0.

Nice isn't the word I'd use. Math and numbers are a creation of man to give regulation to amounts.

What algorithm are you referencing? The one based on multiplication?

Let's look at multiplication and why 6x0=0

If 1x6=6, we can assume 6=1x6. If 2x6=12, 12=2x6
Because 1x6=6, 2x6=6+6, 3x6=6+6+6, etc.
That is because y×6 is defined by how many times it is multiplied. Without having itself at least once, in the case of 0×6, there isn't anything there on the first place.
However, when you look at division, you start with 6, and divide it into x groups.
Regardless if you divide with 2, 3, 1, 6, 0, etc, you always have an amount of 6 within it, just separated.
E.g., I have six o's

oooooo

I want those os to be divided by two

<ooo> <ooo>

Each of two groups has 3, therefore 3.

I have 6 os

oooooo

I want those os to be divided by three

<oo><oo><oo>

Each of three groups has 2, therefore 3

I have 6 os

oooooo

I want those os to be divided by 0

oooooo

Each of those 0 groups has 0. However I have os ungrouped. Therefore I have 6 os.