Math Problem Statement
Say that we have positive integers a and b and that there exist integers x and y so that ax + by = 1. Show that (a, b) = 1.
Solution
We are tasked with proving that if and are positive integers, and there exist integers and such that
then the greatest common divisor .
Proof:
The equation is a linear Diophantine equation. The key point here is that for such an equation to have integer solutions and , the greatest common divisor of and must divide the constant term (which is 1 in this case). Let's analyze this in detail.
- Assume : By definition, is the greatest integer that divides both and . Therefore, we can write
for some integers and , where .
- Substitute and into the original equation: Substituting these expressions into gives
This simplifies to
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Conclude that : The left-hand side is a multiple of , so for the equation to hold true, must divide 1. Since is a positive integer, the only possibility is .
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Conclusion: Therefore, , which means that and are relatively prime.
Thus, we have shown that if there exist integers and such that , then , i.e., and are relatively prime.
Would you like further clarification or additional steps on any part of the proof? Here are some related questions to explore:
- How does the Euclidean algorithm help in finding ?
- What are some practical applications of linear Diophantine equations?
- How do we find integer solutions for ?
- How can we extend this idea to solve other linear Diophantine equations with more variables?
- What is the relationship between the Extended Euclidean Algorithm and solving ?
Tip: The Extended Euclidean Algorithm is an efficient way to find integer solutions and for , as well as computing the greatest common divisor.
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Math Problem Analysis
Mathematical Concepts
Number Theory
Linear Diophantine Equations
Greatest Common Divisor
Formulas
ax + by = 1
gcd(a, b) = d
a = d * a1
b = d * b1
Theorems
Bézout's Identity
Euclidean Algorithm
Suitable Grade Level
Grades 10-12
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