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.

In the words of Euclid:

If two numbers be prime to one another, the sum will also be prime to each of them; and if the sum of two numbers be prime to any one them, the original numbers will also be prime to one another.

(The Elements: Book $\text{VII}$: Proposition $28$)

Proof

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 $a$.

$\Box$

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 $b$.

$\blacksquare$

Historical Note

This proof is Proposition $28$ of Book $\text{VII}$ of Euclid's The Elements.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{VII}$. Propositions