Problem 23 - Fundamentals of Mathematics

Question

Find the greatest common factor (GCF) of the numbers. \(24,30,\) and 45

Step-by-Step Solution

Answer

The greatest common factor of 24, 30, and 45 is 3.

Step 1: List Factors of Each Number

To find the greatest common factor (GCF), we first list out the factors of each number. - Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 - Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30 - Factors of 45: 1, 3, 5, 9, 15, 45

Step 2: Identify Common Factors

Next, identify the common factors from the lists. - Common factors of 24, 30, and 45: 1, 3

Step 3: Select the Greatest Common Factor

Now, choose the largest number from the common factors. - The greatest common factor is 3 because it is the largest number that appears in all lists.

Key Concepts

Factors

Factors are numbers that can be multiplied together to produce another number. For any given number, its factors are integers that divide that number completely without leaving a remainder. Understanding the concept of factors helps in simplifying mathematical problems.

For example, to find the factors of 24, you find all pairs of numbers that multiply to 24, such as 1 and 24, 2 and 12, 3 and 8, and so on. Similarly, for 30 and 45, list all such pairs:

• Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
• Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
• Factors of 45: 1, 3, 5, 9, 15, 45

This foundational step of listing factors is crucial for further analysis in finding common factors and ultimately deducing the greatest common factor.

Common Factors

Once we have the factors for each number, the next task is to identify the factors that are common, meaning the same, in all lists. This concept is pivotal for finding the Greatest Common Factor (GCF) because it aims to find the most extensive factor shared by all numbers.

For the numbers 24, 30, and 45, let's compare their factors listed previously to identify the shared factors:

• Common factors of 24, 30, and 45: 1, 3

These numbers 1 and 3 appear in each of the factor lists. They make up the intersection of factors between these numbers, and are known as the common factors.

The challenge here is not just identifying them but understanding why they matter: having common factors allows us to seamlessly reduce fractions, simplify ratios, and solve equations.

Prime Factorization

Now, let's explore another approach to find common factors through prime factorization. Prime factorization involves breaking down a number into its basic building blocks, which are prime numbers.

For example, instead of listing factors, you can break each number down into its prime factors as follows:

• Prime factorization of 24: \(2^3 \times 3\)
• Prime factorization of 30: \(2 \times 3 \times 5\)
• Prime factorization of 45: \(3^2 \times 5\)

To find the GCF using prime factorization, identify the lowest power of all primes common to each number. From the example, the number 3 is the only prime factor common to all, with the lowest power being 3 itself.

Understanding prime factorization helps in dissecting complex numbers and provides a deeper insight into the innate structure of numbers, improving problem-solving accuracy and efficiency.