Problem 62 - Algebra 2

Question

Find the greatest common factor of each set of numbers. $$ 24,84 $$

Step-by-Step Solution

Answer

The greatest common factor of 24 and 84 is 12.

Step 1: Understand the Problem

The task is to find the greatest common factor (GCF) of the numbers 24 and 84. The GCF of two numbers is the largest positive integer that divides each of the numbers without leaving a remainder.

Step 2: List the Factors of Each Number

To begin, list all the factors of each number: For 24: The factors are 1, 2, 3, 4, 6, 8, 12, and 24. For 84: The factors are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84.

Step 3: Identify the Common Factors

Next, identify the factors that 24 and 84 have in common: The common factors are 1, 2, 3, 4, 6, and 12.

Step 4: Find the Greatest Common Factor

From the list of common factors, determine the largest number. The greatest common factor is the largest number that appears in both lists of factors. The greatest common factor of 24 and 84 is 12.

Key Concepts

Factors

Factors are numbers that divide another number completely, without leaving any remainder. When you hear someone mention factors of a number, they're speaking about all the whole numbers that it can be divided by evenly. For example:

• The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
• This means that each of these numbers divides 24 without leaving a remainder.

Similarly, the factors of 84 include 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84. Finding factors is an essential step when solving many math problems, especially those that involve divisibility or require finding common factors. By listing factors, it becomes easier to identify the relationships between numbers.

Common Factor

A common factor is a factor that two or more numbers share. To find these, first list out all the factors of the numbers in question. Then, look for numbers that appear in both lists. For example, for 24 and 84:

• Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
• Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84

The common factors in both these lists are 1, 2, 3, 4, 6, and 12. Common factors are fundamental when simplifying fractions or finding the greatest common factor (GCF) of two or more numbers. They help in reducing expressions to their simplest forms, making calculations more straightforward.

Divisibility

Divisibility refers to the ability of one number to be divided by another without a remainder. When a number is divisible by another, it implies that it is a multiple of that number. To determine if a number is a factor of another, check if the larger number can be divided by the smaller number evenly—that is, without leaving a leftover or a remainder. For instance:

• 24 is divisible by 12 because when you divide 24 by 12, the result is 2 with no remainder.
• Similarly, 84 is divisible by 12 because 84 divided by 12 equals 7, again with no remainder.

Understanding divisibility is vital for solving problems involving factors and common factors. It's a basic yet crucial concept that forms the foundation of more complex mathematical operations, making calculations simpler and clearer.