In this article, we will explore the GCF of 108 in detail, providing step-by-step instructions and practical examples to ensure you grasp this important mathematical concept. Whether you’re a student preparing for exams or simply looking to enhance your math skills, this guide will serve as your go-to resource.
What is the Greatest Common Factor (GCF)?
The Greatest Common Factor (GCF) of two or more integers is the largest number that divides each of the integers without leaving a remainder. In simpler terms, it’s the biggest number that all the given numbers share as a factor.
Finding the GCF of 108
To determine the GCF of 108, we need to consider the context—whether we’re finding the GCF of 108 with another number or multiple numbers. For this guide, we’ll explore multiple methods to find the GCF of 108 with different numbers, ensuring a comprehensive understanding.
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Method 1: Listing All Factors
One straightforward way to find the GCF is by listing all the factors of each number and identifying the largest common one.
Step-by-Step Example: Finding the GCF of 108 and 36
- List the factors of 108:
- 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108
- List the factors of 36:
- 1, 2, 3, 4, 6, 9, 12, 18, 36
- Identify the common factors:
- 1, 2, 3, 4, 6, 9, 12, 18, 36
- Determine the greatest common factor:
- 36 is the largest common factor.
Thus, the GCF of 108 and 36 is 36.
Method 2: Prime Factorization
Prime factorization involves breaking down each number into its prime factors. The GCF is then found by multiplying the smallest powers of common prime factors.
Step-by-Step Example: Finding the GCF of 108 and 84
- Prime factorize each number:
- 108:
- 108 ÷ 2 = 54
- 54 ÷ 2 = 27
- 27 ÷ 3 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1
- Prime factors: (22*3*3)
- 84:
- 84 ÷ 2 = 42
- 42 ÷ 2 = 21
- 21 ÷ 3 = 7
- 7 ÷ 7 = 1
- Prime factors: (22*3*7)
- Identify common prime factors:
- Both have (2^2) and (3).
- Multiply the smallest powers of common prime factors:
- (2^2 \times 3 = 4 \times 3 = 12)
Thus, the GCF of 108 and 84 is 12.
Method 3: Euclidean Algorithm
The Euclidean Algorithm is a more efficient method, especially for larger numbers. It involves repeated division to find the GCF.
Step-by-Step Example: Finding the GCF of 108 and 30
- Divide the larger number by the smaller number:
- 108 ÷ 30 = 3 with a remainder of 18.
- Replace the larger number with the smaller number and the smaller number with the remainder:
- Now, find the GCF of 30 and 18.
- Repeat the process:
- 30 ÷ 18 = 1 with a remainder of 12.
- 18 ÷ 12 = 1 with a remainder of 6.
- 12 ÷ 6 = 2 with a remainder of 0.
- When the remainder is 0, the divisor is the GCF:
- 6 is the GCF of 108 and 30.
Applying GCF of 108 in Real-Life Scenarios
Understanding the GCF of 108 can be applied in various real-life situations, such as:
- Simplifying Fractions: Simplifying fractions like ( 108/36 ) by dividing both numerator and denominator by their GCF.
- Resource Allocation: Distributing 108 items equally among groups without any leftovers.
- Solving Word Problems: Problems involving ratios and proportions often require finding the GCF for simplification.
Common Mistakes to Avoid
When calculating the GCF of 108, students often make the following mistakes:
- Missing Common Factors: Ensure all factors are listed accurately when using the listing method.
- Incorrect Prime Factorization: Double-check each step to ensure prime factors are correctly identified.
- Misapplying the Euclidean Algorithm: Carefully follow each division step to avoid calculation errors.
Practice Problems
To reinforce your understanding, try solving these practice problems:
- Find the GCF of 108 and 24 using the listing method.
- Determine the GCF of 108 and 90 using prime factorization.
- Use the Euclidean Algorithm to find the GCF of 108 and 42.
Answers:
- GCF of 108 and 24 is 12.
- GCF of 108 and 90 is 18.
- GCF of 108 and 42 is 6.