# Zero Divides Zero

## Theorem

Let $n \in \Z$ be an integer.

Then:

$0 \divides n \implies n = 0$

That is, zero is the only integer divisible by zero.

## Proof

 $\ds 0$ $\divides$ $\ds n$ $\ds \leadsto \ \$ $\ds \exists q \in \Z: \,$ $\ds n$ $=$ $\ds q \times 0$ Definition of Divisor of Integer $\ds \leadsto \ \$ $\ds n$ $=$ $\ds 0$ Integers have no zero divisors, as Integers form Integral Domain.

$\blacksquare$