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 aa and bb are positive integers, and there exist integers xx and yy such that

ax+by=1,ax + by = 1,

then the greatest common divisor (a,b)=1(a, b) = 1.

Proof:

The equation ax+by=1ax + by = 1 is a linear Diophantine equation. The key point here is that for such an equation to have integer solutions xx and yy, the greatest common divisor of aa and bb must divide the constant term (which is 1 in this case). Let's analyze this in detail.

  1. Assume d=gcd(a,b)d = \gcd(a, b): By definition, dd is the greatest integer that divides both aa and bb. Therefore, we can write

a=da1andb=db1a = d \cdot a_1 \quad \text{and} \quad b = d \cdot b_1

for some integers a1a_1 and b1b_1, where gcd(a1,b1)=1\gcd(a_1, b_1) = 1.

  1. Substitute a=da1a = d \cdot a_1 and b=db1b = d \cdot b_1 into the original equation: Substituting these expressions into ax+by=1ax + by = 1 gives

da1x+db1y=1.d \cdot a_1 x + d \cdot b_1 y = 1.

This simplifies to

d(a1x+b1y)=1.d \cdot (a_1 x + b_1 y) = 1.

  1. Conclude that d=1d = 1: The left-hand side is a multiple of dd, so for the equation to hold true, dd must divide 1. Since dd is a positive integer, the only possibility is d=1d = 1.

  2. Conclusion: Therefore, gcd(a,b)=d=1\gcd(a, b) = d = 1, which means that aa and bb are relatively prime.

Thus, we have shown that if there exist integers xx and yy such that ax+by=1ax + by = 1, then (a,b)=1(a, b) = 1, i.e., aa and bb are relatively prime.


Would you like further clarification or additional steps on any part of the proof? Here are some related questions to explore:

  1. How does the Euclidean algorithm help in finding gcd(a,b)\gcd(a, b)?
  2. What are some practical applications of linear Diophantine equations?
  3. How do we find integer solutions for ax+by=1ax + by = 1?
  4. How can we extend this idea to solve other linear Diophantine equations with more variables?
  5. What is the relationship between the Extended Euclidean Algorithm and solving ax+by=1ax + by = 1?

Tip: The Extended Euclidean Algorithm is an efficient way to find integer solutions xx and yy for ax+by=1ax + by = 1, 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