The division method — also known as the ladder method or cake method — is the fastest way to calculate the Least Common Multiple of two or more numbers on paper. Use the quick navigation below to jump to any section:
The division method — also popularly known as the ladder method or cake method — is one of the fastest and most reliable techniques for finding the LCM of two or more numbers. Instead of listing out endless multiples or factoring every single number into long trees separately, you divide the numbers together in a single grid using common prime factors until they can no longer be divided.
This method is especially handy for LCMs of three or more numbers, where listing multiples would take forever. Let's break it down step by step.
The method gets its two famous alternative names from the way we write it down on paper:
Both names refer to the exact same technique. Different textbooks and curricula across the US, UK, Canada, India, and Australia use different names, so don't be confused if you hear either — they both produce the identical result.
Here is the universal 4-step algorithm you can use on any set of numbers:
Write all the given numbers horizontally separated by commas, with an L-shaped vertical division bracket on the left.
Find the smallest prime number (2, 3, 5, 7, etc.) that divides at least two of the numbers. Divide and write the quotients below.
Keep repeating step 2 until no prime divides any two of the remaining numbers in the current row.
Multiply all the divisors on the left column together with all leftover numbers in the bottom row. That gives your LCM!
Let's find LCM(12, 18) using the division method. This is the classic foundational example you'll encounter everywhere in middle school math.
12, 18
The ladder method really shines when you have three or more numbers. Listing multiples of 4, 6, and 8 takes valuable test time, but the ladder gets the answer in 20 seconds.
2, 3, 4, 2 and 4 are both divisible by 2. The 3 is not divisible by 2, so carry 3 straight down:1, 3, 2. No prime divides two or more of these numbers, so stop here.Let's try another classic multi-number challenge: LCM(9, 15, 30).
3, 5, 10, 5 and 10 are divisible by 5. Carry 3 down:3, 1, 2, no prime divides two or more numbers. Stop here!
How does the ladder / cake method stack up against other ways of calculating LCM?
| Method | Best Used For | Speed | Difficulty |
|---|---|---|---|
| Listing Multiples | Very small numbers (e.g., 2 and 3) | Very slow for large numbers | Very Easy |
| Prime Factorization | Two numbers, or when prime factor trees are required | Medium | Moderate |
| Division / Ladder Method | Three or more numbers, or any pair | Fastest on paper | Easy once practiced |
| GCD Formula (Product ÷ GCD) | Two large integers when GCD is known | Fastest with calculator | Easy |
These are the most frequent errors students make when using the division method on homework and exams:
| ❌ Common Mistake | Why It Causes Errors | ✅ Correct Technique |
|---|---|---|
| Using composite divisors (e.g. 4, 6, 9) | Non-primes can hide remaining shared factors in the bottom row | Always use strictly prime numbers: 2, 3, 5, 7, 11, etc. |
| Dividing when only 1 number has that factor | Unnecessarily inflates the product if done carelessly | Only divide if at least two numbers share the prime factor |
| Forgetting to multiply the bottom row | Gives only the common divisors (closer to GCD than LCM) | Multiply all left-side divisors AND the final bottom row |
| Dropping numbers that don't divide | Loses the number completely from the calculation | Always carry undivided numbers straight down to the next row |
| Jumping over small primes | Skipping 2 or 3 to divide by 5 can make numbers messy | Always test primes in order: start with 2, then 3, then 5, etc. |
Try these practice problems on paper using the ladder method, then verify your results with the answer key below:
| Problem | Ladder Working (Divisors × Bottom Row) | LCM Answer |
|---|---|---|
| LCM(6, 8) | 2 × (3 × 4) = 24 | 24 |
| LCM(10, 15) | 5 × (2 × 3) = 30 | 30 |
| LCM(4, 6, 8) | 2 × 2 × (1 × 3 × 2) = 24 | 24 |
| LCM(12, 16, 24) | 2 × 2 × 2 × 3 × (1 × 2 × 1) = 48 | 48 |
| LCM(8, 12, 20) | 2 × 2 × (2 × 3 × 5) = 120 | 120 |
| LCM(15, 20, 30) | 5 × 3 × 2 × (1 × 2 × 1) = 60 | 60 |
While every method has its strengths, the ladder method is especially powerful in specific scenarios:
1/6 + 1/8 + 1/10 is fastest with the ladder.The division method — also called the ladder or cake method — is one of the most practical tools in school mathematics. Here is the summary in 3 rules:
The division method (also called the ladder or cake method) finds the LCM by repeatedly dividing the given numbers by common prime factors. You then multiply all the divisors along with the final remainders to get the LCM.
Using the ladder method: divide by 2 → (6, 9); divide by 3 → (2, 3); stop. LCM = 2 × 3 × 2 × 3 = 36.
Yes. The ladder method, cake method, and division method all refer to the same technique for finding LCM and GCD. Different textbooks use different names for the same thing.
Yes. Using non-prime divisors (like 4 or 6) can leave hidden common factors in the bottom row, giving the wrong answer. Always use prime divisors: 2, 3, 5, 7, 11, and so on.
Then that divisor should not be used. You only divide when at least two numbers in the row share the prime factor. The number that doesn't divide stays unchanged and is carried down to the next row.
Yes. Once you finish the ladder, the GCD is simply the product of the divisors on the left side that divided all numbers. For LCM, you also multiply the final bottom row. So both values come from the same ladder.
Absolutely. The ladder works for any number of inputs (3, 4, 5, or more). Just keep dividing the entire row by prime factors that divide at least two numbers, until no more divisions are possible.