# Why zero is a multiple of every integer, but not a divisor of zero?

All positive and negative numbers including zero are called integers. So in the form $a=bq$, since $0 = 0ㆍq$ is true for any integer $q$, $0$ can have $0$ as a divisor of itself as well as a multiple of itself by the definition expressed by $a=bq$.

But why it's said "We cannot divide by $0$"? It's understood as "$0$ cannot be a divisor" to me.

"Definition: An integer a is called a multiple of an integer $b$ if $a=bq$ for some integer $q$. In this case we also say that b is a divisor of $a$, and we use the notation $b | a$ . . . On the other hand, for any integer $a$, we have $0 = aㆍ0$ and thus $0$ is a multiple of any integer."

Source: Abstract Algebra: Third Edition, John A. Beachy, William D. Blair, p.4.

"Rule Division by $0$ is undefined. Any expression with a divisor of $0$ is undefined. We cannot divide by $0$"

Source: Prealgebra: A Text/Workbook, Charles McKeague, p.61.

"Observe that division by the integer $0$ is not defined, since for $n≠0$ there is no integer $x$ such that $0ㆍx=n$ and since for $n =0$ every integer $x$ satisfies $0ㆍx=0$"

Source: Introduction to Mathematical Proofs, Second Edition, Charles Roberts, p.99.

[Now I understand my question more after reading number theory chapter of a book]

$0=d\cdot 0$
Thus, 0 is a multiple of every integer except 0.

• Is there a single unique value that zero divided by zero would be? Dec 28, 2015 at 15:49
• @JB King Should there be a single value of zero divided by zero so that zero can be a divisor? Dec 28, 2015 at 17:10
• @buzzee, for all other integer divisions the answer $q$ is a unique value. To have a multi-valued solution does present some challenges compared to other operations. Dec 28, 2015 at 17:31
• You are mixing terminology from three different books with very different focuses. One of them is pre-algebra - intended for students who may be 12 years old. Another is abstract algebra for college students. The underlying phenomenon is always the same, but the terminology that the books use to describe it varies depending on their audience and on the author's taste. Dec 29, 2015 at 14:51
• What do you mean by "$0\cdot q=0$ is not allowed"? We certainly are allowed to write $0\cdot q=0$ for any integer $q$, and this is how we know that $0$ is a multiple of $q$ for any integer $q$. It is also how we know that $0$ is a multiple of $0$ and that $0$ is a divisor of $0$. Jan 4, 2016 at 5:47

Every integer divides zero, including zero itself; however, the only integer that zero divides is itself. That is, $b \mid 0$ for all integers $b$; but if $a$ is an integer and $0 \mid a$, then $a = 0$.
When it is said that "you can't divide by zero", what is meant is that, given an integer $a$, there is not a unique quotient upon division by zero.
Specifically, given integers $a$ and $b$, with $b \ne 0$, if $b \mid a$ then there is a unique integer $q$ such that $a = qb$, namely $q=\frac{a}{b}$. However, if we allow the case when $b=0$, then we lose the uniqueness. Indeed, as already mentioned, $0 = q \cdot 0$ for all integers $q$, so it makes no sense to assign a value to the expression $\frac{0}{0}$. And if $a \ne 0$ then there is no integer $q$ such that $a = q \cdot 0$, so it also doesn't make any sense to assign a value to the expression $\frac{a}{0}$.
• @buzzee It's not wrong. It's just that "$a$ is divisor of $b$" and "$b$ can be divided by $a$" mean two different things. Dec 28, 2015 at 17:39
• @buzzee We define $a/b$ to be the unique number $q$ such that $a=bq$, if it exists. For $b=0$, either existence or uniqueness fails. Dec 28, 2015 at 18:03
• @buzzee It is. The problem is with other direction. If we know that $a$ is divisor of $b$, then we can't conclude $a/b$ exists. For example, take $a=b=0$. $0$ is a divisor of itself, but there is no unique number $q$ giving $0=0q$. Hence $0/0$ doesn't exist. Dec 28, 2015 at 18:20