Ladder Method – Prime Factorization by Repeated Division
The ladder method (also called the division ladder or upside-down division) is a systematic way to find prime factorization. Instead of branching outward like a tree, it stacks divisions downward — making it easy to see and tidy to write.
How the Ladder Method Works
- Write the number inside a partial box (the “ladder rungs”).
- Divide by the smallest prime that goes in: 2, then 3, then 5, then 7...
- Write the quotient below.
- Repeat until the quotient is 1 (or a prime for single-number use).
- The primes on the left are the prime factorization.
Example 1 – Prime Factorization of 72
72 ÷ 2 = 36 | 36 ÷ 2 = 18 | 18 ÷ 2 = 9 | 9 ÷ 3 = 3 | 3 is prime. Primes: 2, 2, 2, 3, 3 → 72 = 2³ × 3²
Example 2 – GCF of 48 and 60 Using Shared Ladder
Divide both numbers by common primes simultaneously. Stop when they share no more common prime divisors.
GCF(48, 60)
Both even → ÷2: 24, 30. Both even → ÷2: 12, 15. Not both even. Both divisible by 3? 12÷3=4; 15÷3=5. Yes → ÷3: 4, 5. No common prime factor left.
GCF = 2 × 2 × 3 = 12. LCM = 12 × 4 × 5 = 240.
Ladder vs Tree – Comparison
| Feature | Factor Tree | Ladder Method |
|---|---|---|
| Layout | Branches outward | Stacks downward |
| Best for | Visual learners | Systematic/written work |
| GCF/LCM together | Harder | Easier with shared ladder |
| Large numbers | Tree gets unwieldy | Clean column layout |
Key Takeaways
- Always divide by the smallest prime possible at each step.
- The ladder method keeps work organised in a column.
- Using a shared ladder lets you find GCF and LCM in one process.
- Both tree and ladder give identical prime factorizations.
Practice Questions
- Use the ladder method to find the prime factorization of 120.
- Use the shared ladder to find GCF(36, 54).
- Find LCM(24, 36) using the ladder method.
- Use the ladder method for 1,260.
- Find GCF and LCM of 90 and 126 using the shared ladder.