The Bézout's theorem and its corollary

The Bézout's theorem

Bézout's theorem tells us that:

Corollary of the Bézout's theorem

Bézout's theorem corollary tells us that::

é

As well, for a simple product:

And it is possible to generalize it for any product of integers:

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Demonstration

The Bézout's theorem

Let be two integers.

1. From left to right implication

Let us assume that and are coprime.

In other words,

We know from the Bézout's identity that:

Since we have assumed that , as a result we get:




2. Reciprocal

Conversely, let now assume that:

Let consider , a common divisor of and .

We know from the properties of divisibility that:

being a common divisor of and , it divides , as well as all the linear combinations of and .

Well, , so .

The only number which divides is itself, then .

It is the only common divisor of and , so .

Thus,




3. Equivalence

And as a result of both implications,

é

Corollary of the Bézout's theorem

Let be three integers.

1. From left to right implication

If , then with the Bézout's theorem, we do have this:

As a result we do notice that:

And finally,

2. Reciprocal

If we do have this:

Then, still with the Bézout's theorem,

Performing the product , we do have this:

Thus,




3. Equivalence

And as a result of both implications, we do obtain the following equivalence:

é




4. Product of two numbers

1. From left to right implication

If we now take the hypothesis where , then again with the Bézout's theorem, we do have:

Now, we can apply the theorem four times from this equivalence and:

As a result we obtain,




2. Reciprocal

If we retrieve the previous ipplication and we try to verify its reciprocal, we do startfrom this hypothesis:

By applying the Bézout's theorem, we do have this:

Indeed, by performing the product of these four expressions, we do obtain:

Now, when distributing all this, we do obtain a kind of tree, when each term will contain either , nor :

such as, after having gathered all these terms according to or :




3. Equivalence

And as a result of both implications, we do obtain the following equivalence:




5. Generalization for any product

By applying the previous reasoning, but for any product of integers, we can generalize this corollary by claiming that: