Coprime Integers cannot Both be Zero
Jump to navigation
Jump to search
Theorem
Let $a$ and $b$ be integers.
Let $a$ and $b$ be coprime.
Then it cannot be the case that $a = b = 0$.
Proof
Let $a$ and $b$ be coprime.
Then by definition:
- $\gcd \set {a, b} = 1$
Aiming for a contradiction, suppose $a = b = 0$.
Then $\gcd \set {a, b}$ is undefined.
But it is not possible both:
- for $\gcd \set {a, b}$ to be undefined
- for $\gcd \set {a, b} = 1$.
Hence by Proof by Contradiction it follows that it cannot be the case that $a = b = 0$.
$\blacksquare$