The Gauss' theorem and its corollary

The Gauss' theorem

Gauss' theorem tells us that :

$$ a / bc \enspace et \enspace a \wedge b = 1 \hspace{0.2em} \Longrightarrow \hspace{0.2em} a/c \qquad (Gauss) $$

Corollary of the Gauss' theorem

Gauss' theorem corollary tells us that :

$$ a / c \enspace et \enspace b / c, \enspace with \enspace a \wedge b = 1 \hspace{0.2em} \Longrightarrow \hspace{0.2em} ab/c \qquad \bigl(Gauss \enspace (corollary)\bigr) $$

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Demonstration

The Gauss' theorem

Let \((a, b, c) \in (\mathbb{Z})^3\) be three integers.

Let us assume that \( a / bc \) and also \( a \) and \( b \) are coprime.

If: \( \Biggl \{ \begin{gather*} a / bc \\ a \wedge b = 1 \end{gather*}\)

Then,

$$ \exists (k, u, v) \in (\mathbb{Z})^3, \Biggl \{ \begin{gather*} bc = ka \qquad (1) \\ au + bv = 1 \qquad (2) \end{gather*} $$

With \( (2) \), we do have this:

However, thanks to \( (1) \), we know that \( bc = ka \), therefore:

We then have \( a / c \).

And finally,

$$ a / bc \enspace et \enspace a \wedge b = 1 \hspace{0.2em} \Longrightarrow \hspace{0.2em} a/c \qquad (Gauss) $$

Corollary of the Gauss' theorem

Let \((a, b, c) \in (\mathbb{Z})^3\) be three integers.

Let us assume that \( a / c \), \( b / c \) and also \( a \) and \( b \) are coprime.

If: \(\Biggl \{ \begin{gather*} a / c \\ b / c \end{gather*}\)

Then,

$$ \exists (k, k') \in \hspace{0.05em}\mathbb{Z}^2, \Biggl \{ \begin{gather*} c = ka \qquad (1) \\ c = k'b \qquad (2) \end{gather*} $$

Hence, from \( (1) \) and \( (2) \) we find that:

We then get \( b/ak \), but \( a \wedge b = 1 \).

From which we can conclude that \( b/k \), that is to say:

But \( c = ka \), so,

We finally realize that \( ab / c \).

And finally,

$$ a / c \enspace et \enspace b / c, \enspace with \enspace a \wedge b = 1 \hspace{0.2em} \Longrightarrow \hspace{0.2em} ab/c \qquad \bigl(Gauss \enspace (corollary)\bigr) $$