Introduction

Relevance of the Topic

Divisors and Multiples, when studied, inaugurate the understanding of the structure of integers. They are fundamental stones in the construction of the mathematical edifice and are crucial for understanding calculations, operations, and algebra.

The ability to identify, calculate, and manipulate divisors and multiples allows students to solve problems in various areas of everyday life and in more advanced fields of mathematics. Such tools are essential for the study of future concepts, such as fractions, prime numbers, factorization, and basic arithmetic operations.

Contextualization

Within the curriculum, the topic "Divisors and Multiples" is located at the core of introductory topics to mathematics in the 6th grade. This topic plays a crucial role in solidifying the fundamental arithmetic skills of students, providing a solid foundation for future mathematical studies. It is also a direct precursor to understanding more advanced topics, such as the study of fractions, equations, and geometry.

The discussion of these concepts allows the visualization of mathematics not as a maze of disconnected numbers, but as an interconnected and cohesive system. Each number has a story to tell and a role to play. A deep understanding of divisors and multiples gives students the power to explore and unravel the intricate world of numbers.

Theoretical Development

Components

• Divisors: A number (A) is a divisor of a number (B) if the division (B/A) leaves no remainder. For example, 3 is a divisor of 15, as 15/3 = 5 and there is no remainder.

  • Proper Divisors: These are the divisors of a number without considering itself and 1. For example, the proper divisors of 12 are 2, 3, and 4.
  • Common Divisors: These are the divisors that two or more numbers have in common. For example, the common divisors of 15 and 20 are 1 and 5.

• Multiples: A number (A) is a multiple of another number (B) if the multiplication (A \cdot B) results in a third number. For example, 24 is a multiple of 6, as 6 \cdot 4 = 24.

  • Least Common Multiple (LCM): It is the smallest number that is a multiple of two or more numbers. For example, the LCM of 4 and 6 is 12, as both have 12 as one of their multiples.
  • Common Multiples: These are the multiples that two or more numbers have in common. For example, the common multiples of 3 and 4 are 0 (zero), 12, 24, 36, etc.

Key Terms

• Dividend Number: It is the number that will be divided into equal parts. In the case of the division (B/A), the number (B) is the dividend.

• Divisor Number: It is the number by which the dividend will be divided. In the case of the division (B/A), the number (A) is the divisor.

• Quotient: It is the result of the division. In the example, 5 is the quotient of 15/3.

• Remainder: Something is left after the division, that is, it was not possible to divide the number into exact parts. In the example of 15/3, there is no remainder.

Examples and Cases

• For the number 8, its proper divisors are 1, 2, and 4, and its multiples are 0 (zero), 8, 16, 24, etc.

• For the numbers 6 and 9, their common divisors are 1 and 3, and their common multiples are 0 (zero), 18, 36, 54, etc. The LCM of 6 and 9 is 18, as it is the smallest number that is a multiple of both.

• For the numbers 15 and 25, there are no common divisors beyond 1, and their common multiples are 0 (zero), 75, 150, 225, etc. It is not possible to calculate the LCM in this case, as there are no multiples beyond zero.

Detailed Summary

Relevant Points

• Definition of Divisors: Every number that, when divided by a second number, results in an exact division without remainder, is called a divisor of the second number. This concept is critical for understanding integers.

• Proper Divisors: These are the divisors that a number has, excluding the number itself and 1. This exclusion makes the study more focused and relevant for specific analyses.

• Common Divisors: If two or more numbers have the same divisor, that number is called a common divisor. This is valuable for finding "links" between different numbers.

• Definition of Multiples: If, by multiplying a number by a second number, we obtain a third number, the first number is called a multiple of the second. This is a fun notion, where students realize that numbers can behave in different ways.

• Least Common Multiple (LCM): It is the smallest number that is simultaneously a multiple of two or more numbers. This concept is useful for simplifying calculations and for solving practical problems.

• Common Multiples: These are the multiples that two or more numbers have in common. This notion connects the idea of multiples with that of divisors, showing that there is a set of numbers that can be divided by each of the original numbers.

Conclusions

• Articulation of Concepts: The interconnection between divisors and multiples is an important aspect. Both share the idea of "sharing" and are linked by the notion of quotient.

• Generalization of Division and Multiplication: The study of divisors and multiples allows students to generalize the concepts of division and multiplication, seeing them as strategies for solving actions beyond calculating results.

• Applicability Beyond Mathematics: These concepts are not limited to mathematics, they have applications in other disciplines, such as science and engineering, and even in everyday life.

Proposed Exercises

1. Common Divisors and Multiples of Number: State five common divisors and three common multiples of 12 and 18.

2. Finding Proper Divisors: List the proper divisors of the numbers 20, 36, and 45.

3. Least Common Multiple: Calculate the LCM of 3 and 4. Use the strategy you prefer.

4. Application of LCM: A music festival will be held every 5 days and a food fair every 7 days. In how many days will the festival and fair be scheduled together again? Use the LCM to solve.