Bézout's theorem: senior high school math course in PDF.

Unclassified

Report an error / Note?

Sommaire de cette fiche

1 I.Statement of Bézout’s theorem:
2 II. a new characterization of the PGCD

Bézout’s theorem in a course of arithmetic for the students of terminale speciality.

I.Statement of Bézout’s theorem:

Theorem:

Let a and b be two non-zero natural numbers.

To say that a and b are prime is equivalent to saying that there are two integers u and v such that $au\,+\,bv\,=\,1$ .

Demonstration:

1.suppose there are two integers u and v such that au + bv = 1 and prove that a and b are prime to each other.

We note $\Delta\,=PGCD(a,b)$

$\Delta$ divides a and b so $\Delta$ divides au + bv.

Since au + bv = 1, $\Delta$ = 1 and a and b are prime to each other.

2.suppose that a and b are prime and show that 1 is written in the form au + bv.

Let $\varphi$ be the set of numbers in the form au + bv with $u\,\in\,\mathbb{Z}$ and $v\,\in\,\mathbb{Z}$ .

The set $\varphi$ is not empty because for u = 1 and v = 0, $a\in\varphi$ .

Thus $\varphi$ contains strictly positive integers, and, among them, one smaller than all the others.

Note this smallest element.

The euclidean division of a by m is written a=mq+r with $0\leq\,\,r<m$

or .

Thus $r\in\varphi$ .Now m is the smallest strictly positive integer of $\varphi$ so r = 0.

Thus m divides a.We show in the same way that m divides b.

Since a and b are prime, m=1 and .

In practice, how to find u and v?
To determine the coefficients, we use Euclid’s algorithm.

Let’s give an example.

We look for a pair (x;y) of relative integers such that 89x+41y=1 (1).

89 and 41 are prime to each other so there are two relative integers x and y satisfying (1).

Let a=89 and b=41.

$89=41\times \,2+7$ so $7=89-2\times \,41=a-2b$ .

$41=7\times \,5+\,6$ so $6=41-7\times \,5=b-5(a-2b)=11b-5a$ .

$7=6\times \,1\,+1$ so .

Let $89\times \,6\,+41\times (-13)=1$ .

Thus is a solution of (1).

II. a new characterization of the PGCD

Theorem:

a and b are two non-zero natural numbers.To say that $\Delta$ is the is equivalent to saying that $\Delta$ is a divisor of a and b and there are two relative integers u and v such that $\Delta\,=au+bv$ .

Cette publication est également disponible en : Français (French) Español (Spanish) العربية (Arabic)

Download and print this document in PDF for free

You have the possibility to download then print this document for free «bézout's theorem: senior high school math course in PDF.» in PDF format.

Other forms similar to bézout's theorem: senior high school math course in PDF..

Bézout's theorem: senior high school math course in PDF.
100
Bézout's theorem in a course of arithmetic for the students of terminale speciality. I.Statement of Bézout's theorem: Theorem: Let a and b be two non-zero natural numbers. To say that a and b are prime is equivalent to saying that there are two integers u and v such that .…
Barycenter: 1st grade math exercises with answers in PDF.
96
Exercises on the barycenter in 1st grade with the use of the definition of the barycenter of n weighted points and properties of the barycenter such as associativity. All of these first grade exercises have detailed answer keys so that students can review online. Exercise 1 - Barycenter of weighted…
Barycenter: 1st grade math exercises with answers in PDF.
96
Exercises on the barycenter in 1st grade with the use of the definition of the barycenter of n weighted points and properties of the barycenter such as associativity. All of these first grade exercises have detailed answer keys so that students can review online. Exercise 1 - Barycenter of weighted…

Les dernières fiches mises à jour.

Voici les dernières ressources similaires à bézout's theorem: senior high school math course in PDF. mis à jour sur Mathovore (des cours, exercices, des contrôles et autres), rédigées par notre équipe d'enseignants.

On Mathovore, there is 13 703 930 math lessons and exercises downloaded in PDF.

Bézout’s theorem: senior high school math course in PDF.

I.Statement of Bézout’s theorem:

II. a new characterization of the PGCD

Other documents in the category Unclassified

Other forms similar to bézout's theorem: senior high school math course in PDF..