Problem 68 - Fundamentals of Mathematics

Question

Find the greatest common factor of each collection of numbers. \(6,8,\) and 12

Step-by-Step Solution

Answer

The greatest common factor of 6, 8, and 12 is 2.

Step 1: Prime Factorization

Find the prime factorization of each number:- 6: The prime factors are 2 and 3 because \(6 = 2 \times 3\)- 8: The prime factors are 2, 2, and 2 because \(8 = 2 \times 2 \times 2\)- 12: The prime factors are 2, 2, and 3 because \(12 = 2 \times 2 \times 3\)

Step 2: Identify Common Factors

Identify the prime factors that are common to each set of numbers. For 6, 8, and 12, the common prime factor is 2.

Step 3: Determine the Greatest Common Factor

The greatest common factor (GCF) is the highest power of the common prime factor that appears in all factorizations. - The prime number 2 appears to the power of 1 in all sets of factorizations. - Therefore, the GCF is 2.

Key Concepts

Prime Factorization

Understanding prime factorization is crucial for solving many math problems, including finding the greatest common factor (GCF). Prime factorization involves breaking down a number into its basic building blocks—prime numbers.

Prime numbers are numbers greater than 1 that have no divisors besides 1 and themselves. For example, 2, 3, 5, and 7 are all prime numbers. To perform prime factorization, you repeatedly divide the number by its smallest prime factor, continuing this process with each quotient until you reach 1.

Let’s consider the number 12. You start by dividing by 2, the smallest prime factor, to get 6. Continuing with 6, divide by 2 again to get 3. Finally, divide by 3 to get 1. This gives us the prime factors: 2, 2, and 3, so we write it as:
\[12 = 2 \times 2 \times 3\]
Breaking numbers down in this manner not only simplifies the process of finding common factors but also aids in a variety of other mathematical procedures.

Common Factors

Common factors are factors that two or more numbers share. After performing prime factorization, identifying these shared factors becomes more straightforward.

In our example, we found the prime factors for 6, 8, and 12 to be:

• 6: 2, 3
• 8: 2, 2, 2
• 12: 2, 2, 3

The next step is to look for the numbers that appear in all three sets. The number 2 is present in the factorization of 6, 8, and 12, making it a common factor.

Identifying common factors is useful not only in calculating the greatest common factor but also in simplifying fractions and solving problems that involve ratios and proportions.

Mathematical Problem-Solving

Mathematical problem-solving is an essential skill, and finding the greatest common factor exemplifies the process. By breaking down complex problems into manageable steps, you enhance your problem-solving toolbox.

The problem-solving steps here include:

• Prime factorization of each number
• Identification of common factors
• Determination of the greatest common factor by considering the highest occurrence of each common factor

In our given problem, after determining the prime factors, we identified that 2 is the only common factor. We then verified its presence in all, resulting in the GCF of 2.

Understanding and applying these steps ensures clarity and efficiency in tackling similar math problems. It improves both competency and confidence in mathematical reasoning.