As developers of mathematical tools, we understand that finding the Greatest Common Factor (GCF) of two numbers can be challenging for many users. Today, we’ll explore how to determine the GCF of 192 and 72 using three different methods: Prime Factorization, Listing Factors, and the Euclidean Algorithm. By the end of this lesson, you’ll be equipped with multiple strategies to tackle similar problems with confidence.
What is the Greatest Common Factor (GCF)?
The Greatest Common Factor (GCF) of two or more numbers is the largest number that divides each of the numbers without leaving a remainder. In other words, it’s the biggest number that both numbers share as a factor.
Example:
For the numbers 8 and 12, the factors of 8 are 1, 2, 4, 8, and the factors of 12 are 1, 2, 3, 4, 6, 12. The common factors are 1, 2, 4, so the GCF is 4.
Methods to Find the GCF of 192 and 72
Below, we provide three different methods to find the GCF. You can also determine the GCF using our free online GCF Calculator
Method 1: Prime Factorization
Prime factorization involves breaking down each number into its prime components. Once we have the prime factors, the GCF is the product of the smallest powers of all common primes.
Step 1: Prime Factorization of Each Number
Factorizing 192:
- 192 ÷ 2 = 96
- 96 ÷ 2 = 48
- 48 ÷ 2 = 24
- 24 ÷ 2 = 12
- 12 ÷ 2 = 6
- 6 ÷ 2 = 3
- 3 ÷ 3 = 1
Factorizing 72:
- 72 ÷ 2 = 36
- 36 ÷ 2 = 18
- 18 ÷ 2 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1
Step 2: Identify Common Prime Factors
- Common primes: 2 and 3
- Smallest exponents:
- For 2: min(6, 3) = 3
- For 3: min(1, 2) = 1
Step 3: Calculate the GCF
GCF=23×31= 8×3=24
Therefore, the GCF of 192 and 72 is 24.
Method 2: Listing All Factors
This method involves listing all factors of each number and identifying the largest common one.
Step 1: List the Factors
- Factors of 192: 1,2,3,4,6,8,12,16,24,32,48,64,96,192
- Factors of 72: 1,2,3,4,6,8,9,12,18,24,36,72
Step 2: Identify Common Factors
- Common factors: 1, 2, 3, 4, 6, 8, 12, 24
Step 3: Determine the GCF
- The largest common factor is 24.
Therefore, the GCF of 192 and 72 is 24.
Method 3: The Euclidean Algorithm
The Euclidean Algorithm is an efficient method for finding the GCF of two numbers, especially large ones. It uses a series of division steps based on the principle that the GCF of two numbers also divides their difference.
Step 1: Apply the Euclidean Algorithm
Given numbers: a=192, b=72
Step 1: 192÷72 = 2 with a remainder of 48 ⇒ 192=72×2+48
Step 2: 72÷48 = 1 with a remainder of 24 ⇒ 72 = 48×1+24
Step 3: 48÷24 = 2 with a remainder of 0 ⇒ 48 = 24×2+0
Step 2: Identify the GCF
- The algorithm stops when the remainder is 0.
- The last non-zero remainder is 24, which is the GCF.
Therefore, the GCF of 192 and 72 is 24.
Conclusion
Regardless of the method you choose, the Greatest Common Factor of 192 and 72 is 24. Here’s a quick recap of the three methods:
- Prime Factorization: Break down each number into prime factors and multiply the smallest powers of common primes.
- Listing Factors: List all factors of each number and identify the largest one they share.
- Euclidean Algorithm: Use division steps to find the last non-zero remainder.
Each method has its own advantages. Prime Factorization is great for understanding the underlying structure of numbers, Listing Factors is straightforward but can be time-consuming for large numbers, and the Euclidean Algorithm is the most efficient for large or multiple numbers.
Practice Problem:
Find the GCF of 84 and 30 using all three methods discussed above and tell us in the comment section
Additional Tips
- Prime Factorization is particularly useful when dealing with multiple numbers, as it allows you to identify all common prime factors simultaneously.
- The Euclidean Algorithm is highly efficient for larger numbers and is widely used in computational applications.
- Understanding all three methods enriches your number sense and provides flexibility in problem-solving.