Math Problem Statement

Let n ∈ ℕ. We define a = 2n + 1 and b = 9n + 4. 1. Determine if 9a - 2b = 1. 2. Deduce that gcd(a, b) = 1.

Solution

The image contains a math exercise in French. Here's the translation and breakdown of the exercise:


Exercise 26:
Let nNn \in \mathbb{N} (set of natural numbers). We define: a=2n+1andb=9n+4a = 2n + 1 \quad \text{and} \quad b = 9n + 4

  1. Determine if: 9a2b=19a - 2b = 1

  2. Deduce that: gcd(a,b)=1\gcd(a, b) = 1


Solution Outline:

  1. For Part 1:

    We have the given values for aa and bb: a=2n+1,b=9n+4a = 2n + 1, \quad b = 9n + 4

    Substitute aa and bb into the equation 9a2b9a - 2b: 9a2b=9(2n+1)2(9n+4)9a - 2b = 9(2n + 1) - 2(9n + 4) Simplify both terms: 9(2n+1)=18n+9,2(9n+4)=18n+89(2n + 1) = 18n + 9, \quad 2(9n + 4) = 18n + 8 Therefore: 9a2b=(18n+9)(18n+8)=19a - 2b = (18n + 9) - (18n + 8) = 1 Hence, the equation holds true.

  2. For Part 2:

    The task is to deduce that gcd(a,b)=1\gcd(a, b) = 1, which means aa and bb are relatively prime.

    The equation 9a2b=19a - 2b = 1 is a form of the Bézout identity. Since the linear combination of aa and bb gives 1, it implies that the greatest common divisor (gcd) of aa and bb is 1. Hence, gcd(a,b)=1\gcd(a, b) = 1.


Would you like more details or have any questions on this? Here are 5 related questions for further exploration:

  1. Can you generalize the solution to any linear combination xayb=1xa - yb = 1?
  2. How can Bézout’s identity help in proving the gcd of two numbers?
  3. How does the gcd help in solving Diophantine equations like 9a2b=19a - 2b = 1?
  4. What other methods can we use to compute gcd(a,b)\gcd(a, b) apart from Bézout’s identity?
  5. Can this approach be extended to numbers involving different coefficients, like 5a3b=15a - 3b = 1?

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