# 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. Mar 6, 2017 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. Mar 6, 2017 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. Mar 6, 2017 at 15:32
• @Joffan But that k could be anything. So everything divides 0 right ? that is the opposite of what cited in the question.
– user312097
Mar 6, 2017 at 15:34
• Mar 6, 2017 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 $x$ and $y$ are two integers then $x$ is said to be divisible by $y$ if there exists an integer value $p$ such that $x = py$