Date: 10/21/2005 at 15:38:11
From: Doctor Vogler
Subject: Re: factors of zero

Hi Joseph,

Thanks for writing to Dr. Math. There are two common definitions for
"factors" of a number. One is natural numbers, and the other is
integers.

If you are only considering natural numbers, then you ask: What are
the natural numbers (factors) that can be multiplied by other natural
numbers (corresponding factors) to get n? For the natural number 12,
those are the ones you listed:

1, 2, 3, 4, 6, and 12.

But if you are considering integers, then you ask: What are the
integers (factors) that can be multiplied by other integers
(corresponding factors) to get n? For the integer 12, those are:

1, 2, 3, 4, 6, 12, -1, -2, -3, -4, -6, and -12.

You asked about the factors of 0. In a similar way, you could ask
about the factors of -5. In either case, these are not natural
numbers, so it doesn't make sense to use the first definition of a
factor. But if we use the second definition of factor, then the
factors of -5 are -5, -1, 1, and 5 (which are also the factors of 5),
and every integer is a factor of zero.

In fact, zero is the only integer that has infinitely many factors,
and zero is the only integer that has zero for a factor.

Finally, I should point out one more thing: I usually interpret
"natural number" to mean a positive integer, and many mathematicians
mean exactly this. But some mathematicians use "natural number" to
mean a nonnegative integer. So that means that zero is also a natural
number. If you use that definition for a natural number, then the
first definition of factor makes sense for zero (since zero is now a
natural number) and its factors are all natural numbers (including zero).

If you have any questions about this or need more help, please write
back, and I will try to clear things up.

- Doctor Vogler, The Math Forum
https://mathforum.org/dr.math/