Is Zero a Factor of Zero?

Inspiration for this post came from an unlikely source: a bottom set year seven class and a non-specialist maths teacher. He was teaching them factors, and was asked by a student, "is zero a factor of zero?"

This question would have just about made me explode with excitement. I think my colleague's reaction was a bit different. He is a humanities teacher by trade, so naturally he was very good at fobbing them off with an answer that sounded impressive without actually answering the question.

This colleague has obviously been working with me for too long now, because after this lesson he came and asked me the same question, because he was actually interested in the answer. I even caught him reading my post Is Zero a Square Number?  at lunchtime. He's one of us now.

So, is zero a factor of zero? Well, as is often the case in maths, the answer is: it is if you want it to be.

It all depends on how you define a factor. Here are some possible ways.
For n and m natural numbers, n is a factor of m if:
1) n divides m with no remainder.
2) n x p = m, where p is a natural number.

[Side note: obviously natural numbers have negative factors too, and factors can be defined on integers rather than just the natural numbers, but negative factors aren't interesting, they're just the same as the positive ones but with a minus sign.]

Let's look at the two definitions:

1) From this definition, for zero to be a factor of zero, zero would have to divide zero with no remainder. What is zero divided by zero? That's my all-time favourite maths debate (and you all know I love to maths debate). Here are three possible answers:

-Anything divided by zero is infinity, therefore the answer is infinity.
-Zero divided by anything is zero, therefore the answer is zero.
-Anything divided by itself is one, therefore the answer is one.

In fact the answer could be anything you want it to be:
0 x pi = 0
Therefore 0/0 = pi.

So we say the question (and therefore the answer) is undefined. Or "MA ERROR" on your old Casio.

So by this definition, zero is not a factor of zero, in fact it can't be a factor of anything. However, every other number must be a factor of zero. Zero divided by anything other than zero is zero, which is a whole number with no remainder.

2) I think we'd all agree there exists a p such that 0 x p = 0. There are infinitely many such p! So by this definition, zero is a factor of zero.

So what is the answer? Well I'm going to solve this mathematical mystery the way mathematicians solve most of the really puzzling mathematical mysteries. I'm going to use the magic words: "by convention".

By convention, zero is not a factor of itself.

Done.

Emma x x x