Question:** What is the greatest common factor of 72 and 108? - Decision Point
What Is the Greatest Common Factor of 72 and 108? A Clear Explanation
What Is the Greatest Common Factor of 72 and 108? A Clear Explanation
When working with numbers in math, one key concept students frequently encounter is the greatest common factor (GCF), also known as the greatest common divisor (GCD). If you’ve ever asked, “What is the greatest common factor of 72 and 108?”—you’re on the right track to mastering this essential concept. In this article, we’ll explore the GCF of 72 and 108 in detail, including step-by-step calculation methods and why understanding this number matters in math.
Understanding the Context
Understanding the Greatest Common Factor (GCF)
The greatest common factor of two or more integers is the largest positive integer that divides all of them without leaving a remainder. For example, the GCF of 72 and 108 is the largest number that evenly divides both 72 and 108.
How to Find the GCF of 72 and 108
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Key Insights
There are three common methods to calculate the GCF:
- Prime Factorization
- Listing Factors
- Using the Euclidean Algorithm
1. Prime Factorization Method
Break each number into its prime factors:
- 72 = 2³ × 3²
- 108 = 2² × 3³
Now, identify the lowest exponent of each common prime factor:
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- Common primes: 2 and 3
- Smallest exponent of 2: 2¹ (or 2², taking the minimum of 2² and 2³ → 2²)
- Smallest exponent of 3: 3²
Multiply these together:
GCF = 2² × 3² = 4 × 9 = 36
2. Listing Factors (Suitable for Smaller Numbers)
List all factors of each number:
- Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
- Factors of 108: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108
The largest number appearing in both lists is 36.
3. Euclidean Algorithm (Efficient for Large Numbers)
The Euclidean algorithm uses repeated division. Here’s how it works:
- Divide the larger number by the smaller: 108 ÷ 72 = 1 with remainder 36
- Now divide 72 by 36: 72 ÷ 36 = 2 with remainder 0
- When the remainder is 0, the last non-zero remainder is the GCF
So, GCF(72, 108) = 36
