Integer Combination of Coprime Integers/Sufficient Condition/Proof 3

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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 $1$:

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


Let $a \perp b$.

Thus they are not both $0$.

Let $S$ be defined as:

$S = \set {\lambda a + \mu b: \lambda, \mu \in \Z}$

$S$ contains at least one strictly positive integer, because for example:

$a \in S$ (setting $\lambda = 1$ and $\mu = 0$)
$b \in S$ (setting $\lambda = 0$ and $\mu = 1$)

By Set of Integers Bounded Below has Smallest Element, let $d$ be the smallest element of $S$ which is strictly positive.

Let $d = \alpha a + \beta b$.

Let $c \in S$, such that $\lambda_0 a + \mu_0 b = c$ for some $\lambda_0, \mu_0 \in \Z$.

By the Division Algorithm:

$\exists \gamma, \delta \in \Z: c = \gamma d + \delta$

where $0 \le \delta < d$


\(\ds \delta\) \(=\) \(\ds c - \gamma d\)
\(\ds \) \(=\) \(\ds \paren {\lambda_0 a + \mu_0 b} - \gamma \paren {\alpha a + \beta b}\)
\(\ds \) \(=\) \(\ds \paren {\lambda_0 - \gamma \alpha} a + \paren {\mu_0 - \gamma \beta} b\)
\(\ds \) \(\in\) \(\ds S\)

But we have that $0 \le \delta < d$.

We have defined $d$ as the smallest element of $S$ which is strictly positive

Hence it follows that $\delta$ cannot therefore be strictly positive itself.

Hence $\delta = 0$ and so $c = \gamma d$.

That is:

$d \divides c$

and so the smallest element of $S$ which is strictly positive is a divisor of both $a$ and $b$.

But $a$ and $b$ are coprime.

Thus it follows that, as $d \divides 1$:

$d = 1$

and so, by definition of $S$:

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


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.