Numbers are Coprime iff Sum is Coprime to Both

Theorem
Let $a, b$ be integers.

Then:
 * $a \perp b \iff a \perp \paren {a + b}$

where $a \perp b$ denotes that $a$ and $b$ are coprime.

Necessary Condition
Let $a \perp b$.

Suppose $a + b$ is not coprime to $a$.

Then:
 * $\exists d \in \Z_{>1}: d \divides a, d \divides \paren {a + b}$

But then:
 * $d \divides \paren {\paren {a + b} - a}$

and so:
 * $d \divides b$

and so $a$ and $b$ are not coprime.

From this contradiction it follows that $a + b$ is coprime to $a$.

Sufficient Condition
Let $a + b$ be coprime to $a$.

Suppose $a$ is not coprime to $b$.

Then:
 * $\exists d \in \Z_{>1}: d \divides a, d \divides b$

and so:
 * $d \divides \paren {a + b}$

and so $a$ and $\paren {a + b}$ are not coprime.

From this contradiction it follows that $a$ is coprime to $b$.