Introduction à la Factorisation des Nombres Premiers

Study Notes

Prime factorization is the process of expressing a composite number as a product of its prime factors.
A prime number is a whole number greater than 1 that has only two factors: 1 and itself. Examples include 2, 3, 5, 7, 11, etc.
A composite number is a whole number greater than 1 that has more than two factors.
Every positive integer greater than 1 can be expressed uniquely, except for the order of the factors, as a product of prime numbers.

Identify the smallest prime number: Begin by dividing the given number by the smallest prime number (2).
Divisibility Check: If the number is divisible by the prime, divide. Otherwise, move to the next prime number.
Repeated Division: Continue this process, using the same prime factor or moving to the next prime, until the quotient is 1.
Express as Product: Finally, write the original number as a product of the prime factors obtained, using exponents for repeated factors.

Find the prime factorization of 36.
- 36 is divisible by 2. 36 ÷ 2 = 18
- 18 is divisible by 2. 18 ÷ 2 = 9
- 9 is divisible by 3. 9 ÷ 3 = 3
- 3 is a prime number. 3 ÷ 3 = 1
- The prime factorization of 36 is 2 x 2 x 3 x 3 or 2² x 3².

Tree Diagram: A visual method. Repeatedly break down the number into factors until all factors are prime.
Ladder/Division Method: A systematic approach using repeated division by prime factors. Prime factors are written vertically, outside the division bracket.
Trial and Error: Attempt division by increasing prime numbers until a factor is found that results in an integer result.

Finding the Greatest Common Factor (GCF): The GCF of two or more numbers is the largest factor they share. Found by identifying common prime factors and multiplying them together, using the smallest exponent for each common factor.
Finding the Least Common Multiple (LCM): The LCM of two or more numbers is the smallest multiple they share. Found by identifying all prime factors appearing in the factorizations and multiplying them together, using the largest exponent for each factor.
Simplifying Fractions: Used to determine if a fraction can be simplified by identifying common factors in the numerator and denominator.
Cryptography: Crucial in certain cryptographic algorithms for key generation and interpretation. The difficulty in factoring large numbers strengthens security protocols.
Number Theory: A fundamental concept in number theory, which studies properties of numbers, primarily prime numbers.