Divisibility relation on nonnegative integers.

Question

Definition: $a$ divides $b$ if $b = ak$ for some integer $k$.

The book says that it is reflexive. But what about $0/0$ ? I am missing any point ?

Is it not a paradox since for standard division operation it is not defined?

Answer 1

The definition of divisibility states that $a|b$ if their exists a number $k$ such that $ka=b$. Let $b=0$ and let $a$ be any integer (positive or negetive). Then $a|0$ since their is some number $k$ (what $k$ will work) such that $ka=0$.

EDIT to address the comment The division operation $(a,b)\mapsto\frac{a}{b}$ asks for a unique $k$ such that $a=kb$. But if $a=b=0$, then any $k$ will do.

Answer 2

By that definition, $0 = a\cdot0$, therefore every number divides 0. I don't see any problem.