Factors of Zero

Date: 10/21/2005 at 14:38:53
From: Joseph
Subject: factors of zero

I need to make a list of the factors of zero.  It is easy for natural
numbers like 12; {1,2,3,4,6,12}, but the list for zero stumps me.

All natural numbers divide evenly into zero, so the set should be the
set of natural numbers.  But each natural number would need to be
multiplied by zero in order to produce zero, so zero should be
included in the set of factors.
 
However, I cannot divide zero by zero, therefore I cannot include zero
in its set of factors.  I am puzzled by this contradiction.

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
  http://mathforum.org/dr.math/