Problem 1 - Fundamentals of Mathematics

Question

Find the GCF of the following numbers. 24 and 36

Step-by-Step Solution

Answer

The GCF of 24 and 36 is 12.

Step 1: List Factors of Each Number

First, we need to list out all the factors of each number. For 24, the factors are: 1, 2, 3, 4, 6, 8, 12, and 24. For 36, the factors are: 1, 2, 3, 4, 6, 9, 12, 18, and 36.

Step 2: Identify Common Factors

Next, we find the numbers that are common in both lists of factors. The common factors of 24 and 36 are: 1, 2, 3, 4, 6, and 12.

Step 3: Determine the Greatest Common Factor (GCF)

From the list of common factors, identify the greatest (largest) factor. The greatest common factor of 24 and 36 is 12.

Key Concepts

Understanding the Concept of Factors

When attempting to find the greatest common factor (GCF) of two numbers, it is crucial to first understand what factors are. Factors are the building blocks of numbers. They are integers that can be multiplied together to form another integer. For example, the factors of 24 include numbers that, when multiplied in pairs, give 24. These pairs can be 1 and 24, 2 and 12, 3 and 8, 4 and 6. Each number in these pairs is a factor of 24.

To determine the factors of a number, you repeatedly divide the number by integers, starting from 1, and continuing through the smallest prime numbers until the quotient is less than the divisor. This systematic listing of numbers leads you to identify all the combinations that represent the factors.

Identifying Common Factors

Once you have listed out the factors of each number, the next step is to find which of these numbers appear in both lists. These shared numbers are known as common factors.

For example, when examining the factors of 24 and 36, both numbers share common divisors. In the lists, the numbers 1, 2, 3, 4, 6, and 12 are present for both.

• Common factors help to find relationships between numbers, especially when dealing with calculations involving multiple integers.
• They simplify expressions and equations where these numbers are involved.

Recognizing common factors is a critical skill in mathematics, as it sets the foundation for further operations like simplifying fractions or solving higher-level algebraic expressions.

Techniques for Listing Factors

With the task of listing factors, familiarity with some efficient techniques will save time and effort. Here's a structured approach to finding factors:

• Begin with the number 1 and the given number itself, as they are universally factors of any integer.
• Divide the number by successive integers, increasing incrementally. Check if the quotient is a whole number. If it is, document both the divisor and the quotient as factors.
• Continue this process up to the square root of the original number because any factor larger than its square root will have been paired with a smaller factor within this range.

Utilizing this method allows you to systematically define the complete set of factors. This approach not only aids in achieving precision but also in facilitating an understanding of the factorization process. By mastering these techniques, one can more readily solve problems involving the GCF or other factor-related queries.