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

Unclassified Report an error on this Mathovore page.Report an error / Note?
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 m=au_1+bv_1 this smallest element.

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

or r=a-mq=a-(au_1+bv_1)q=a(1-u_1q)+b(-v_1q).

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 au_1+bv_1=1.

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 1=7-6=a-2b-11b+5a=6a-13b.

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

Thus (x_0;y_0)=(6;-13) 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 PGCD(a,b) 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 documents in the category Unclassified


Download our free apps with all corrected lessons and exercises.

Application Mathovore sur Google Play Store.    Application Mathovore sur Apple Store.     Suivez-nous sur YouTube.

Other forms similar to 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 .…
  • 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…
  • 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.

  1. Abonnements
  2. Maths : cours et exercices corrigés à télécharger en PDF.
  3. Subscriptions
  4. Suscripciones
  5. الاشتراكات

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

Mathovore

FREE
VIEW