Exercice 2 (Arithmétique dans Z) [00257]

Introduction

This tutorial focuses on the mathematical concept of divisibility within the integers, as explored in the video "Exercice 2 (Arithmétique dans Z)." Understanding divisibility is crucial for solving various arithmetic problems and is a foundational concept in number theory. This guide will break down the key points and exercises from the video to help reinforce your knowledge and application of divisibility.

Step 1: Understanding Divisibility

• Definition: A number ( a ) is said to be divisible by another number ( b ) if there exists an integer ( k ) such that ( a = b \times k ).
• Notation: This is often expressed as ( b \mid a ) (read as "b divides a").
• Examples:
  • ( 15 \mid 45 ) because ( 45 = 15 \times 3 ).
  • ( 7 \nmid 20 ) because there is no integer ( k ) such that ( 20 = 7 \times k ).

Practical Tips

• To check if ( b ) divides ( a ), perform the division ( a \div b ) and see if the result is an integer (no remainder).
• Common pitfalls include overlooking negative numbers and zero, which can affect divisibility.

Step 2: Exploring Common Divisors

• Definition: A common divisor of two integers is a number that divides both numbers without leaving a remainder.
• Finding Common Divisors:
  1. List the divisors of each number.
  2. Identify the common elements from these lists.

Example

• For numbers 12 and 18:
  • Divisors of 12: 1, 2, 3, 4, 6, 12
  • Divisors of 18: 1, 2, 3, 6, 9, 18
  • Common divisors: 1, 2, 3, 6

Step 3: Evaluating True or False Statements

• Bonus Exercise: The video presents a true or false question related to divisibility.
• To evaluate such statements:
  • Analyze the statement logically.
  • Use examples or counterexamples to support your conclusion.

Practical Advice

• When unsure, substitute specific values into the statement to check its validity.

Conclusion

Understanding divisibility is essential in arithmetic and number theory. This tutorial covered definitions, how to determine divisibility, finding common divisors, and evaluating true or false statements. To further your learning, practice with various integer pairs and explore more complex divisibility rules, such as those involving prime numbers or using divisibility tests for 2, 3, 5, and 10. For additional exercises and written corrections, visit http://exo7.emath.fr.