https://file.aiquickdraw.com/mathbot/file/1728857132821cawe7q6r.jpg

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

---

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

1. Determine if:
   \[
   9a - 2b = 1
   \]

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

---

### Solution Outline:

1. **For Part 1:**

   We have the given values for \( a \) and \( b \):
   \[
   a = 2n + 1, \quad b = 9n + 4
   \]

   Substitute \( a \) and \( b \) into the equation \( 9a - 2b \):
   \[
   9a - 2b = 9(2n + 1) - 2(9n + 4)
   \]
   Simplify both terms:
   \[
   9(2n + 1) = 18n + 9, \quad 2(9n + 4) = 18n + 8
   \]
   Therefore:
   \[
   9a - 2b = (18n + 9) - (18n + 8) = 1
   \]
   Hence, the equation holds true.

2. **For Part 2:**

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

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

---

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 \( xa - yb = 1 \)?
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 \( 9a - 2b = 1 \)?
4. What other methods can we use to compute \( \gcd(a, b) \) apart from Bézout’s identity?
5. Can this approach be extended to numbers involving different coefficients, like \( 5a - 3b = 1 \)?

### 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.

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.

Explore the solution of the problem where a = 2n + 1 and b = 9n + 4. Determine if 9a - 2b equals 1, and deduce that gcd(a, b) equals 1 using key number theory concepts such as linear combinations and Bézout's Identity. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation