Does zero divide zero

I wanted to know is zero divisible by zero? I've read that division by zero is not allowed in mathematics, but for instance in Apostol's Introduction to analytic number theory, it states that only $0$ divides $0$, and I've seen problems in form $0\mid f(x)$, wanting the possible amounts of $x$ (integer amounts).

This might be similar to some other topics, but they examine the problem in the set of real numbers (analytic sight), but mine is concered with elementary number theory which concerns integers.

• AFAIK, Everything divides 0 except 0. – super saiyan Mar 6 '17 at 15:28
• In arithmetics, one says that $b$ divides $a$ if there is $c$ such that $a = b c$. With this definition, zero divides zero alright. – Andreas Caranti Mar 6 '17 at 15:32
• Can I find a $k$ such that $k\cdot 0=0$? Yes, easily. So zero divides zero. Can I find a $k$ for any other result? No. So zero divides only zero, no other number. – Joffan Mar 6 '17 at 15:32
• @DietrichBurde: This is not a duplicate of that question. – Asaf Karagila Mar 6 '17 at 15:33
• – Bill Dubuque Mar 6 '17 at 19:34

In the context of fields, like the rational numbers or the real numbers, $0$ does not divide anything, since division is given by multiplying by the multiplicative inverse (which exists from the axioms).

However, in the context of the natural numbers we define the divisibility relation as follows: $$n\mid m\iff\exists k:k\cdot n=m.$$

In that case, every number divides $0$, including $0$ itself. (Note that $\exists k$ is bounded to the natural numbers here!)

If you're asking if zero is divisible by zero and you agree on the definition of divisibility to be

If $x$ and $y$ are two integers then $x$ is said to be divisible by $y$ if there exists an integer value $p$ such that $x = py$

Then zero is certainly divisible by zero as every integer follows that definition. If your notion of divisibility actually means to be able to divide those two and get some definite value then it is not divisible But according to what I think it is.