Determine all factors of zero (divisors of $0$)

Question

How many integer factors of $0$ are there, and what are they?

I'm just curious, but what counts as a factor of $0$? My guess is that there are an infinite number of factors of $0$, but is there a proof?

Answer 1

The statement

$a$ is a factor of $b$

means

$b=ka$ for some integer $k$.

Take $b=0$: then no matter what $a$ is, the equation $0=ka$ is always true for some value of $k$, namely, $k=0$. So every integer is a factor of $0$.

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This is an excellent example of the importance of using definitions precisely. Some people think that a factor of $b$ means

$b$ divided by any integer "which goes exactly",

for example, the factors of $10$ are $10/1$, $10/(-2)$, $10/5$ etc. This would give the factors of $0$ as being $0/1$, $0/2$ etc, in other words, $0$ only, which is not correct.

Answer 2

This question is about terminology, and one can only give an answer if you provide the precise definition of "factor" you are using. Curiously the term does not appear to often get a formal definition, contrary to "divisor" which the term more conventionally associated to the divisibility relation. One may then for convenience consider those two terms to be synonyms. However, it should be noted that most of the time "factor" is used in connection with factorisation. For instance it is more common to speak about prime factors than about prime divisors; the two basically mean the same thing, but a given prime factor might be considered to occur multiple times for a same number, while I don't think one would say that for a prime divisor (which is just a divisor of the number that happens to be prime). In this context it is relevant to note that the number $0$ does not have a prime factorisation and is explicitly excluded from the prime factorisation theorem (usually in one sweep with excluding negative numbers and sometimes also the number$~1$; indeed those numbers pose some difficulties too, but these are less fundamental than the problem with$~0$). Therefore talking about factors of $0$ may be considered not very appropriate. Certainly asking how many factors$~17$ the number$~0$ has does not make any sense.

Apart from that, the following remarks can be made

• The phrases "$a$ divides $b$" (written $a\mid b$), "$a$ is a divisor of $b$", and "$b$ is a multiple of $a$" express grosso modo the same relation, but that does not mean they can always be used interchangeably. Notably divisors are often implicitly assumed to be positive (for instance when talking about the number of divisors or the sum of the divisors of a (positive) number, though this is not required for $a$ in $a\mid b$. Also many authors will not say that $0$ divides $0$, presumably because one cannot divide $0$ by $0$ (I think most of them prefer to leave the divisibility of $0$ by $0$ undefined rather than false). I've cited in this answer an example of authors that make their position in this respect very clear.

• I think most people would agree at least that all nonzero numbers divide$~0$, which provides an answer of sorts to your question.

• However in ring theory the term "zero divisor" means something else, and it does not apply to any integer.

Answer 3

Take any non-zero k, it can be written 0=k.0