Page 2: Common Divisors and Greatest Common Divisor (GCD)

This page delves into the concept of common divisors and introduces the Greatest Common Divisor (GCD), along with coprime numbers.

Definition: A common divisor 'd' of numbers 'a' and 'b' is a number that divides both 'a' and 'b' without leaving a remainder.

Example: Finding common divisors of 24 and 34:

• Divisors of 24: 1, 2, 3, 4, 6, 8, 12, 24
• Divisors of 34: 1, 2, 17, 34
• Common divisors: 1 and 2

Highlight: The Greatest Common Divisor (GCD) is the largest number that divides two numbers without leaving a remainder.

Definition: Two numbers are coprime (or relatively prime) when their greatest common divisor is 1.

Example: 14 and 27 are coprime because their GCD is 1.

Page 1: Number Decomposition and Prime Factorization

This page introduces fundamental concepts of number theory and prime factorization. The content focuses on breaking down numbers into their prime factors and finding divisors.

Definition: Prime factorization is the process of decomposing a number into a product of its prime factors.

Example: The number 504 is decomposed as follows: 504 = 2 × 252 504 = 2 × 2 × 126 504 = 2 × 2 × 2 × 63 504 = 2³ × 7 × 3²

Highlight: When finding divisors of a number, systematic decomposition helps identify all possible factors. For instance, 198 has divisors: 1, 2, 3, 6, 9, 11, 18, 22, 33, 66, 99, and 198.

Vocabulary: Euclidean division involves a dividend, divisor, quotient, and remainder, where the remainder must be less than the divisor.