Methods for Finding the GCF / HCF
There are three common methods for finding the GCF. The best choice depends on the size of the numbers and how quickly you need the answer.
Method 1 – Listing Factors
List all factors of each number. Find the common factors. Pick the largest.
Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Common: 1, 2, 3, 6 → GCF = 6
Method 2 – Prime Factorization
Write each number as a product of primes. Multiply the primes that appear in both factorizations (using the lowest power).
36 = 2² × 3² 48 = 2⁴ × 3
Common prime factors: 2 (lowest power = 2²) and 3 (lowest power = 3¹)
GCF = 2² × 3 = 4 × 3 = 12
Method 3 – Ladder (Division) Method
Divide both numbers by any common prime. Keep dividing until no common factor remains. Multiply all the divisors.
Both even → divide by 2: 30, 45. Both divisible by 3: 10, 15. Both divisible by 5: 2, 3. No common factor left.
GCF = 2 × 3 × 5 = 30
Method Comparison
| Method | Best For | Speed |
|---|---|---|
| Listing factors | Small numbers | Slow for large numbers |
| Prime factorization | Medium numbers, multiple numbers | Moderate |
| Ladder/Euclidean | Large numbers, two numbers | Fastest |
Key Takeaways
- All three methods give the same answer — choose based on the numbers.
- Prime factorization is great when you already need the factorization.
- The ladder method is the most efficient for large numbers.
Practice Questions
- Find GCF(30, 42) using the listing method.
- Find GCF(60, 84) using prime factorization.
- Find GCF(120, 180) using the ladder method.
- Find GCF(13, 17). What does this tell you about these numbers?
- Find GCF(126, 210) using any method.