Definition of $b|a \implies 0|0$? The definition I'm using for $b|a$ (taken from Elementary Numbery Theory by Jones & Jones):

If $a,b \in \mathbb{Z}$ then $b$ divides $a$ if for some $q \in \mathbb{Z}$ $a = qb$. 

However, I have $0 = q\cdot0$ for any $q$ I choose. So this seems to imply that $0$ divides $0$ which I know is always taken to be undefined. Should the definition be for "unique $q$" rather than for "some $q$"? 
Thank-you. 

The statement $0$ divides $0$ and the "quantity" $0/0$ are different things. The first is exactly the statement that there exists some $a$ such that $0a=0$ and the second is not a number
