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January 5th, 2019, 10:56 AM  #1 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 124  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, 11:04 AM  #2 
Senior Member Joined: Oct 2009 Posts: 753 Thanks: 261 
A partial converse is, if there are integers A and B such that Am+Bn=1, then gcd(m,n)=1.

January 7th, 2019, 11:01 AM  #3  
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 124  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, 07:49 AM  #4 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 124 
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? 

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bezout, converse, identity 
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