Problem 123 - Fundamentals of Mathematics

Question

(Section 3.5) Find the greatest common factor of 12 and 18 .

Step-by-Step Solution

Answer

The greatest common factor of 12 and 18 is 6.

Step 1: List the Factors of Each Number

First, list all the factors of both 12 and 18. The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 18 are 1, 2, 3, 6, 9, and 18.

Step 2: Identify the Common Factors

Next, identify the common factors of 12 and 18 from the lists. The common factors of 12 and 18 are 1, 2, 3, and 6.

Step 3: Select the Greatest Common Factor

From the common factors identified in Step 2, select the largest number. The greatest common factor of the common factors 1, 2, 3, and 6 is 6.

Key Concepts

Factors

Understanding the concept of factors is crucial in mathematics, especially when solving problems like finding the greatest common factor (GCF). A factor is a number that divides another number completely, leaving no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 because each of these numbers can divide 12 without any leftovers.

To find the factors of any given number:

• Start with the number 1 and the number itself as the first two factors.
• Check each integer in between to see if it divides the number perfectly.
• If it does, note it as a factor.

Finding factors lays the groundwork for determining common factors between two numbers. Practicing this with different numbers helps to strengthen your mathematical problem-solving skills.

Common Factors

Once you've identified the factors of two numbers, the next step is to find the common factors. Common factors are simply the numbers that appear in both sets of factors. For instance, when considering the factors of 12 (1, 2, 3, 4, 6, 12) and 18 (1, 2, 3, 6, 9, 18), you identify the common ones by comparing the two lists.

The common factors here are: 1, 2, 3, and 6. These numbers divide both 12 and 18 without leaving a remainder. Being good at spotting common factors is essential not just for finding the greatest common factor, but also for simplifying fractions, calculating least common multiples, and more.

Practicing with various number pairs can help solidify your understanding.

Mathematics Problem Solving

Problem solving in mathematics often involves a structured approach, which can significantly ease the path to the correct answer. Let's delve into the approach of finding the greatest common factor (GCF) as an instance of this structured problem-solving technique.

When tasked with finding the GCF, follow these steps:

• List out the factors for each number involved.
• Identify the common factors between them.
• Select the largest number from these common factors, as it represents the GCF.

This logical sequence leverages your understanding of factors and common factors to efficiently direct you to the solution. Not only does this method apply to finding the GCF, but it also preps you with problem-solving foundations you can build on in various mathematical scenarios.

Practicing this approach can enhance your analytical skills, enabling you to tackle similar problems with confidence and clarity.