
Abstract Algebra Abstract Algebra Math Forum 
 LinkBack  Thread Tools  Display Modes 
January 5th, 2019, 11:56 AM  #1 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,641 Thanks: 119  Bezout's Identity converse
Bezout's Identity: If d = gcd(m,n), there exist integers A and B st d=Am+Bn. (proved by Euclid's algorithm} 1) Is there a converse? 2) How would you express it? 3) How would you prove it? 
January 5th, 2019, 12:04 PM  #2 
Senior Member Joined: Oct 2009 Posts: 696 Thanks: 235 
A partial converse is, if there are integers A and B such that Am+Bn=1, then gcd(m,n)=1.

January 7th, 2019, 12:01 PM  #3  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,641 Thanks: 119  Quote:
The following theorem is copied from Meserve, Fundamental Concepts of Algebra: "Theorem 211. The greatest common divisor of any two positive integers m and n can be found as the last nonvanishing remainder n$\displaystyle _{k}$ in the Euclidean Algorithm. There exist integers A and B such that (210) (m,n) = n$\displaystyle _{k}$ = Am + Bn" For example: m=22, n=24 22=1x16+6, (22,16)=(16,6) 16=2x6+4, (16,6)=(6,4) 6=1x4+2, (6,4)=(4,2)=2 4=2x2, from which: 6=221x16 4=162x6=162x(221x6)=3x162x22 2=61x4=221x161x(3x162x22)=3x224x16=Ax22+Bx16 In this case n$\displaystyle _{k}$ is 2 Now the question. Again fron Meserve: "The equation (210) is both necessary and sufficient for n$\displaystyle _{k}$ to be the greatest common divisor of m and n. It is necessary because of Theorem 211, and sufficient since if (210) holds, every common factor of m and n divides n$\displaystyle _{k}$." What is this saying, ie, translate it into: If "R" then "S". If "S" then "R". What are "R" and "S"  
January 9th, 2019, 08:49 AM  #4 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,641 Thanks: 119 
If n$\displaystyle _{k}$ is last nonzero remainder in Euclidean Algorithm, then n$\displaystyle _{k}$ is (m,n) and A and B exist st (m,n)=Am+Bn. If n$\displaystyle _{k}$ is (m,n) and A and B exist st (m,n)=Am+Bn, then n$\displaystyle _{k}$ is last nonzero remainder in Euclidean Algorithm. What's the point? n$\displaystyle _{k}$ is unique but A and B aren't. Best I could come up with. Got a better idea? 

Tags 
bezout, converse, identity 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Stating the converse, converse of if x=y then x^2 = y^2?  Tangeton  Algebra  3  April 8th, 2016 09:00 AM 
Circle theorems and their converse  Tangeton  Geometry  1  March 10th, 2016 08:40 PM 
Is there a statement equivalent to its own converse?  RyanPowers  Applied Math  3  September 17th, 2014 09:19 AM 
Set theory. Is the converse true?  omoplata  Applied Math  1  June 11th, 2011 06:59 AM 
Bezout's identity and odd co primes  Euzenius  Number Theory  3  July 16th, 2009 11:41 AM 