Problem 8 - Fundamentals of Mathematics

Question

Find the greatest common factor (GCF) of the numbers. 8 and 12

Step-by-Step Solution

Answer

The GCF of 8 and 12 is 4.

Step 1: List the Factors of Each Number

First, identify all factors of each number. - Factors of 8: 1, 2, 4, 8 - Factors of 12: 1, 2, 3, 4, 6, 12

Step 2: Identify Common Factors

Next, find the factors that both numbers share. Compare the factors of 8 and 12: - Common factors: 1, 2, 4

Step 3: Determine the Greatest Common Factor

Identify the largest number in the list of common factors, which is the greatest common factor. - Greatest common factor: 4

Key Concepts

Understanding Factors

Factors are numbers you multiply together to get another number. They're essentially the building blocks of the number. To find the factors of a number, divide it by all integers up to that number. If the division leaves no remainder, then the divisor is a factor. For example, the factors of 8 include 1 (8 divided by 1 equals 8), 2 (8 divided by 2 equals 4), 4 (8 divided by 4 equals 2), and 8 (itself).
Likewise, numbers bigger than 1 always have at least two factors, 1 and themselves. Thinking of factors helps to break down numbers into simpler components, important in finding commonalities and simplifying problems.

Exploring Common Factors

Common factors are those that two or more numbers share. To find them, first, list all factors for each number, then determine which are present in all lists. For instance, the factors of 8 are 1, 2, 4, and 8, while the factors of 12 are 1, 2, 3, 4, 6, and 12.

• From these lists, identify the numbers that appear in both: 1, 2, and 4.

Using common factors is essential for simplifying fractions and finding the greatest common factor. Recognizing these shared factors helps in bridging numbers together when working with mathematical equations and expressions.

Prime Factorization Insights

Prime factorization involves breaking down a number into its basic building blocks known as prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. To prime factorize a number, repeatedly divide it by the smallest prime until you can't divide anymore.
For instance, to prime factorize 8, divide by 2, the smallest prime number: 8 divided by 2 equals 4, then 4 divided by 2 equals 2, and 2 divided by 2 equals 1. This results in the prime factors: 2, 2, and 2.

• Similarly, the prime factorization of 12 is 2 x 2 x 3.

Prime factorization is a powerful tool for finding the GCF because the GCF can be directly obtained by multiplying the smallest power of each common prime factor. In our example, the GCF of 8 and 12, from the common prime factor 2, would be 4 (since 2 appears twice in the factorization of both numbers).