Cours Arithmétique dans Z ( Partie 1 ) " Divisibilité - Congruence- Division euclidienne "

Introduction

This tutorial covers the foundational concepts of divisibility, congruence, and Euclidean division as introduced in the first part of the arithmetic course in Z. Understanding these concepts is crucial for anyone studying number theory or preparing for advanced mathematics.

Step 1: Understanding Divisibility

Divisibility is a fundamental concept in arithmetic. A number ( a ) is said to be divisible by another number ( b ) if there exists an integer ( k ) such that:

• ( a = b \times k )

Practical Advice

• Check for divisibility by small integers (2, 3, 5, etc.) as a first step.
• Use the divisibility rules:
  • A number is divisible by 2 if its last digit is even.
  • A number is divisible by 3 if the sum of its digits is divisible by 3.
  • A number is divisible by 5 if it ends in 0 or 5.

Common Pitfalls

• Miscalculating when applying divisibility rules can lead to incorrect conclusions.
• Always ensure that ( b ) is not zero, as division by zero is undefined.

Step 2: Exploring Congruence

Congruence is a relationship between two numbers that share the same remainder when divided by a third number. The notation for congruence is:

• ( a \equiv b \mod m )

This means ( a ) and ( b ) leave the same remainder when divided by ( m ).

Practical Advice

• To determine if ( a \equiv b \mod m ), perform the following:
  1. Calculate the remainders of ( a ) and ( b ) when divided by ( m ).
  2. Check if these remainders are equal.

Real-World Application

• Congruences are often used in cryptography, computer science, and coding theory.

Step 3: Applying Euclidean Division

Euclidean division provides a way to divide two integers ( a ) (the dividend) and ( b ) (the divisor) and express the result in a specific form:

• ( a = b \times q + r )

Where:

• ( q ) is the quotient,
• ( r ) is the remainder (where ( 0 \leq r < b )).

Practical Advice

• To apply Euclidean division:
  1. Perform the division to find ( q ) as the integer part.
  2. Calculate ( r ) using the formula above.

Example

If ( a = 17 ) and ( b = 5 ):

• ( q = \lfloor 17 / 5 \rfloor = 3 )
• ( r = 17 - (5 \times 3) = 2 )
• Thus, ( 17 = 5 \times 3 + 2 ).

Conclusion

In this tutorial, we've explored key concepts of divisibility, congruence, and Euclidean division. These foundational principles are essential for further studies in number theory and mathematics.

Next Steps

• Practice problems involving divisibility and congruence.
• Explore more advanced topics in modular arithmetic and its applications.