Can Bezout's Theorem Prove Coprime Relationship in Modular Arithmetic?

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In summary, the conversation involves proving that given the equation d=gdf(a,b) and ax+by=d, x and y are coprime or (x,y)=1. The use of modular arithmetic is being considered, and the hint suggests rewriting a and b as two parts, with one part being "gdf(a,b)" and exploring what happens to "gdf" if (x,y) > 1.
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what I am trying to prove is that given the d=gdf(a,b) and ax+by=d prove that x and y are coprime or i guess (x,y)=1

i don't know whether or not to use modular arithmetic.
 
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twodice said:
what I am trying to prove is that given the d=gdf(a,b) and ax+by=d prove that x and y are coprime or i guess (x,y)=1

i don't know whether or not to use modular arithmetic.


Hint write a and b as two parts each with one part being "gdf(a,b)" What happens to the "gdf" if (x,y) > 1
 

FAQ: Can Bezout's Theorem Prove Coprime Relationship in Modular Arithmetic?

What is Bezout's Theorem?

Bezout's Theorem is a fundamental theorem in algebraic geometry that states that the number of common points on two algebraic curves is equal to the product of their degrees, if the curves do not have any common components.

Who discovered Bezout's Theorem?

Bezout's Theorem is named after the French mathematician Étienne Bézout, who first stated the theorem in 1779.

What is the significance of Bezout's Theorem?

Bezout's Theorem is significant because it allows us to determine the number of solutions to a system of polynomial equations. It is also used in algebraic geometry to study the intersection of curves and surfaces.

Can Bezout's Theorem be applied to higher dimensions?

Yes, Bezout's Theorem can be extended to higher dimensions and can be used to study the intersection of higher-dimensional algebraic varieties.

Is Bezout's Theorem applicable to all types of curves?

No, Bezout's Theorem is only applicable to algebraic curves, which are curves defined by polynomial equations. It cannot be applied to other types of curves, such as transcendental curves.

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