Problem 66 - Fundamentals of Mathematics

Question

Find the greatest common factor of each collection of numbers. 6 and 14

Step-by-Step Solution

Answer

The greatest common factor of 6 and 14 is 2.

Step 1: Understand the Definition

The greatest common factor (GCF) is the largest positive integer that divides two or more numbers without leaving a remainder. It’s also known as the greatest common divisor (GCD).

Step 2: List the Factors of Each Number

List all of the factors for each number. For 6, the factors are: 1, 2, 3, and 6. For 14, the factors are: 1, 2, 7, and 14.

Step 3: Identify the Common Factors

Look at the factors of both numbers and identify the common ones. The numbers that appear in both lists of factors are the common factors. For 6 and 14, the common factors are: 1 and 2.

Step 4: Determine the Greatest Common Factor

From the common factors, choose the largest one. For 6 and 14, the greatest common factor is 2.

Key Concepts

Common Factors

When trying to find the greatest common factor between numbers, it is essential to first determine the common factors. A common factor is a number that evenly divides two or more numbers without leaving any remainder. To find these common factors, we start by listing all the factors of each individual number. For example, to determine the common factors of the numbers 6 and 14:

• List the factors of 6: 1, 2, 3, and 6.
• List the factors of 14: 1, 2, 7, and 14.
• Identify the numbers that appear in both lists.

In this case, the numbers 1 and 2 appear in both lists, making them the common factors. It's a simple yet crucial step that lays the groundwork for determining the greatest common factor, which is the largest number among these common factors.

Divisors

Divisors play a significant role in identifying factors of numbers. A divisor of a given number is a number that divides it completely without leaving any remainder. When listing the factors of a number, essentially you are listing its divisors. Each divisor shows a potential way a number can be evenly divided.
For example:

• For the number 6, the divisors, or factors, are: 1, 2, 3, and 6.
• For the number 14, the divisors, or factors, are: 1, 2, 7, and 14.

To effectively find the greatest common factor, we compare divisors of each number. The common divisors are those that appear in both lists, providing insight into the shared divisible parts between the numbers. Understanding divisors is key in resolving number-based problems and recognizing the mathematical relationships within sets of numbers.

Number Theory

Number theory is a branch of mathematics that deals with the properties and relationships of numbers, especially integers. Within this field lies the study of various important concepts, including factors and divisors.
One of the essential operations in number theory is finding the greatest common factor (GCF), which involves exploring how integers relate through their factors. This concept helps in simplifying fractions, solving equations, and understanding number patterns.

• The GCF is critical in various applications, like reducing fractions to their simplest form.
• It assists in finding common denominators for fractions.
• It even helps in problem-solving for algebraic expressions and equations.

Number theory builds a foundation for understanding complex mathematical concepts by starting with simple principles, like factors and divisors. This powerful yet foundational area of mathematics provides tools for clarifying and solving different mathematical problems, emphasizing the importance of finding greatest common factors.