Problem 2 - Fundamentals of Mathematics

Question

Find the GCF of the following numbers. 48 and 72

Step-by-Step Solution

Answer

The GCF of 48 and 72 is 24.

Step 1: List the Factors of Each Number

First, we need to find all the factors of 48. The factors are numbers that divide 48 without leaving a remainder. The factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. Next, list the factors of 72, which are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.

Step 2: Identify the Common Factors

Now, compare the factors of 48 and 72 to find the common factors. These are the numbers that appear in both lists of factors. The common factors for 48 and 72 are: 1, 2, 3, 4, 6, 8, 12, 24.

Step 3: Find the Greatest Common Factor

Out of all the common factors identified, the greatest one is called the Greatest Common Factor (GCF). Examine the list of common factors: 1, 2, 3, 4, 6, 8, 12, 24. The largest of these numbers is 24. Therefore, the GCF of 48 and 72 is 24.

Key Concepts

Factors

The concept of factors is foundational in understanding how numbers relate to each other. A factor is a number that divides another number completely, leaving no remainder. For example, when we say that 2 is a factor of 48, it means that 48 divided by 2 is a whole number, specifically 24, with no leftovers.

When listing the factors of a number, you start from 1, which is a universal factor for any number, and include all numbers up to (and including) the number itself. For instance, the factors of 48 include:

• 1
• 2
• 3
• 4
• 6
• 8
• 12
• 16
• 24
• 48

Each of these numbers divides 48 exactly.

Understanding factors is key to performing operations like finding the greatest common factor, simplifying fractions, and solving various mathematical problems.

Common Factors

After identifying the factors of two numbers, the next step is to find the common factors. Common factors are numbers that are factors of all the numbers you are considering. Imagine comparing the lists of factors for 48 and 72. Both lists contain:

• 1
• 2
• 3
• 4
• 6
• 8
• 12
• 24

These are the numbers that divide both 48 and 72 exactly.

Common factors offer a shortcut to simplify calculations. For example, if you want to simplify fractions or find relationships between numbers, recognizing their common factors can save time and enhance understanding.

Division in Mathematics

Division is an essential operation in mathematics that helps us understand how numbers distribute into equal parts. When finding factors or common factors, division plays a crucial role.

The process involves taking a number and dividing it by another. If you receive a quotient that's a whole number with no remainder, you've identified a factor. For instance, dividing 48 by 6 gives 8, with no remainder, confirming that 6 is a factor of 48.

This division method enables us to find the greatest common factor (GCF). By comparing the common factors, we see which division results led to the highest whole number that's shared among the integers in question. For 48 and 72, dividing by their GCF, 24, will yield whole numbers in both cases, solidifying a deeper grasp of their numeric relationship.