Zero Divides Zero

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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

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

$\blacksquare$


Sources