GCD: The King of Divisors

Relevance of the Topic

GCD, or Greatest Common Divisor, is a noble figure in mathematics. It is the largest divisor that two or more numbers have in common. Its kingdom is vast, with applications in various fields, including fraction simplification and solving problems of equitable division. It is a key concept that opens the doors to the study of prime numbers, factorization, and equation-solving algorithms. Without the GCD, many mathematical calculations would become complicated and inefficient.

Contextualization

Within the 7th grade Mathematics curriculum, the study of GCD is included in a broader section that addresses integers and their properties. It acts as a bridge between previous topics, such as multiples and divisors, and future topics, such as fractions and proportions. Knowledge of GCD is necessary to understand the algorithms for irreducible fractions and to factor polynomials, which will be addressed later. Therefore, understanding GCD is crucial to advance on the rich path of mathematics!

Theoretical Development

Components

• Common Divisors: To understand the GCD, it is first necessary to master the concept of common divisors. Common divisors of two or more numbers are the divisors that these numbers have in common. They are like the "keys" that fit all the "locks" (the numbers to be divided).

  • Ex: Common divisors of 12 and 18 are 1, 2, 3, 6.

• Greatest Common Divisor: The greatest common divisor (GCD) is the highest value common divisor that two or more numbers have. Essentially, it is the "king" of common divisors.

  • Ex: GCD of 12 and 18 is 6.

• Prime Number Factorization: Factorization is the expression of a number as a product of its prime divisors. The ability to factor prime numbers is fundamental to determining the GCD.

  • Ex: 12 = 2 * 2 * 3, 18 = 2 * 3 * 3. The GCD is the product of the common factors, raised to the lowest exponent: 2 * 3 = 6.

Key Terms

• Division: Mathematical operation that consists of dividing a quantity into equal parts.
• Divisor: Number that divides another number exactly.
• Prime Number: A natural number greater than 1 that cannot be formed by the multiplication of two smaller natural numbers.
• Prime Factor: A prime number that divides another number exactly.

Examples and Cases

• Example 1: Find the GCD of 24 and 36.

  • Common divisors: 1, 2, 3, 4, 6, 12. Maximum: 12.
  • Factorization: 24 = 2 * 2 * 2 * 3, 36 = 2 * 2 * 3 * 3. GCD = 2 * 2 * 3 = 12.

• Example 2: Find the GCD of 15, 25, and 40.

  • Common divisors: 1, 5. Maximum: 5.
  • Factorization: 15 = 3 * 5, 25 = 5 * 5, 40 = 2 * 2 * 2 * 5. GCD = 5.

• Case 1: Simplify the fraction 48/60. Find the GCD of 48 and 60 and divide both numbers by the GCD. Simplified fraction: 4/5.

• Case 2: Divide a pile of 84 candies equally among 6 children. The maximum number of candies each child can have is the GCD of 84 and 6, which is 6. Thus, each child will receive 6 candies.

Detailed Summary

Relevant Points

• The definition of Greatest Common Divisor (GCD), which is the largest common divisor of two or more numbers. This concept is crucial for application in various situations, such as fraction simplification and solving problems of equitable division.
• The term "common divisor", which refers to a number that divides two or more numbers equally. This term is central to understanding the GCD.
• The importance of prime number factorization to determine the GCD. Factorization is the expression of a number as a product of its prime divisors, an essential skill for determining the GCD.
• The technique of determining the GCD through prime number factorization and then identifying the common prime factors of each number, raising them to the lowest exponent and multiplying them.

Conclusions

• The GCD is an essential mathematical tool, with applications in various contexts, including fractions and equitable division.
• Knowledge about prime number factorization is crucial for determining the GCD.
• The ability to identify and manipulate common divisors and common prime factors is vital for applying the concept of GCD.

Exercises

1. Find the GCD of 16 and 24.
2. Determine the GCD of 21, 35, and 42.
3. Simplify the fraction 72/90 by finding the GCD and dividing both numbers by the GCD.