Integer Combination of Coprime Integers

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Theorem

Let $a, b \in \Z$ be integers, not both zero.

Then:

$a$ and $b$ are coprime

if and only if:

there exists an integer combination of them equal to $1$:
$\forall a, b \in \Z: a \perp b \iff \exists m, n \in \Z: m a + n b = 1$

In such an integer combination $m a + n b = 1$, the integers $m$ and $n$ are also coprime.


General Result

Let $a_1, a_2, \ldots, a_n$ be integers.


Then $\gcd \set {a_1, a_2, \ldots, a_n} = 1$ if and only if there exists an integer combination of them equal to $1$:

$\exists m_1, m_2, \ldots, m_n \in \Z: \ds \sum_{k \mathop = 1}^n m_k a_k = 1$


Proof

The proof can conveniently be divided into two parts:

Sufficient Condition

Let $a, b \in \Z$ be integers, not both zero.

Let $a$ and $b$ be coprime to each other.

Then there exists an integer combination of them equal to $1$:

$\forall a, b \in \Z: a \perp b \implies \exists m, n \in \Z: m a + n b = 1$


Necessary Condition

Let $a, b \in \Z$ be integers, not both zero.

Let $a$ and $b$ be such that there exists an integer combination of them equal to $1$.

Then $a$ and $b$ are coprime:

$\forall a, b \in \Z: \exists m, n \in \Z: m a + n b = 1 \implies a \perp b$

In such an integer combination $m a + n b = 1$, the integers $m$ and $n$ are also coprime.


Also known as

This result is sometimes known as Bézout's Identity, although that result is usually used for the more general result for not necessarily coprime integers.

Some sources refer to this result as the Euclidean Algorithm, but the latter as generally understood is the procedure that can be used to establish the values of $m$ and $n$, and for any pair of integers, not necessarily coprime.


Sources

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