Math Problem Statement
Solution
The image contains a math exercise in French. Here's the translation and breakdown of the exercise:
Exercise 26:
Let (set of natural numbers). We define:
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Determine if:
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Deduce that:
Solution Outline:
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For Part 1:
We have the given values for and :
Substitute and into the equation : Simplify both terms: Therefore: Hence, the equation holds true.
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For Part 2:
The task is to deduce that , which means and are relatively prime.
The equation is a form of the Bézout identity. Since the linear combination of and gives 1, it implies that the greatest common divisor (gcd) of and is 1. Hence, .
Would you like more details or have any questions on this? Here are 5 related questions for further exploration:
- Can you generalize the solution to any linear combination ?
- How can Bézout’s identity help in proving the gcd of two numbers?
- How does the gcd help in solving Diophantine equations like ?
- What other methods can we use to compute apart from Bézout’s identity?
- Can this approach be extended to numbers involving different coefficients, like ?
Tip:
In number theory, if a linear combination of two integers equals 1, the integers are coprime, i.e., their gcd is 1. This is a direct consequence of Bézout's theorem.
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Math Problem Analysis
Mathematical Concepts
Number Theory
Linear Combinations
GCD (Greatest Common Divisor)
Formulas
9a - 2b
gcd(a, b)
Bézout's Identity
Theorems
Bézout's Theorem
Greatest Common Divisor (GCD)
Suitable Grade Level
Grades 10-12
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