Zero Divides Zero

From ProofWiki
Jump to navigation Jump to search

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

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

$\blacksquare$


Sources