Problem 69 - Fundamentals of Mathematics

Question

Find the greatest common factor of each collection of numbers. 18 and 24

Step-by-Step Solution

Answer

The greatest common factor of 18 and 24 is 6.

Step 1: Prime Factorization

First, find the prime factorization of both numbers. For 18: Divide by 2: 18 ÷ 2 = 9 Divide by 3: 9 ÷ 3 = 3 Divide by 3: 3 ÷ 3 = 1 So, the prime factors of 18 are 2 × 3 × 3 = 2 × 3^2. For 24: Divide by 2: 24 ÷ 2 = 12 Divide by 2: 12 ÷ 2 = 6 Divide by 2: 6 ÷ 2 = 3 Divide by 3: 3 ÷ 3 = 1 So, the prime factors of 24 are 2 × 2 × 2 × 3 = 2^3 × 3.

Step 2: Identify Common Factors

List out the prime factors for both numbers and identify the common factors. For 18: The prime factors are 2, 3, and 3 again (3^2). For 24: The prime factors are 2, 2 again, 2 again (2^3), and 3. The common prime factors are 2 and 3.

Step 3: Calculate the Greatest Common Factor

To find the greatest common factor (GCF), multiply the lowest power of all the common prime factors. The common prime factors between 18 and 24 are 2 and 3. For 2, the lowest power is 2^1. For 3, the lowest power is 3^1. Thus, the GCF = 2^1 × 3^1 = 2 × 3 = 6.

Key Concepts

Prime Factorization

Prime factorization is a method used to break down a number into its basic building blocks, which are prime numbers. Prime numbers are those numbers greater than 1 that have no divisors other than 1 and themselves. This process involves dividing the number successively by prime numbers until the quotient is 1.
For instance, let's look at the number 18:

• Divide by 2: 18 ÷ 2 = 9
• Divide by 3: 9 ÷ 3 = 3
• Divide by 3 again: 3 ÷ 3 = 1

Thus, the prime factors of 18 are 2 and 3. When multiplied, these factors give us the original number: 2 × 3 × 3 = 18.
Similarly, for 24:

• Divide by 2: 24 ÷ 2 = 12
• Divide by 2 again: 12 ÷ 2 = 6
• Divide by 2 once more: 6 ÷ 2 = 3
• Divide by 3: 3 ÷ 3 = 1

This results in the prime factors 2 and 3: 2 × 2 × 2 × 3 = 24.
Using prime factorization simplifies the process of calculating the greatest common factor by providing focus to the problem.

Common Factors

Common factors are numbers that are shared divisors between two or more numbers. When solving mathematics problems that involve finding the greatest common factor, identifying common factors is essential.
In our example with 18 and 24, once we identify the prime factors:

• 18 has 2 and 3 as prime factors (2 × 3²).
• 24 has 2 and 3 as prime factors (2³ × 3).

The common factors here are 2 and 3. Both numbers share these prime numbers. Common factor identification helps reduce the problem to its core essentials, paving the way to determining the greatest common factor.
By listing and comparing prime factors, we better understand the numerical relationship between numbers.

Mathematics Problem Solving

Mathematics problem solving often involves breaking problems into smaller, more manageable parts. Solving for the greatest common factor is a perfect example of this kind of approach.
Here's how you can apply problem-solving skills in this context:

• Use prime factorization to dissect each number.
• List out all the factors and identify which ones are shared between the numbers.
• Multiply the shared factors using the smallest power of each.

In our example with numbers 18 and 24, after solving each step using effective problem-solving skills, the greatest common factor turns out to be 6.
Mathematics problem-solving encourages a structured approach that not only simplifies problems but also enhances comprehension and application skills in future math challenges.