Factors, Multiples & Primes
Methods for Finding the LCM
Just like the GCF, the LCM can be found by three different methods. Pick the one that suits the numbers and the situation.
Method 3 deserves special attention because of what it reveals: LCM(a, b) × GCF(a, b) always equals a × b, no matter what a and b are. This is not a coincidence — it is a direct algebraic identity, and it means that once you (or a computer) can find the GCF quickly using the Euclidean Algorithm, you get the LCM essentially for free, without ever needing to factor either number into primes at all. That combination — fast GCF, free LCM — is exactly why real software never actually lists multiples the way we do by hand.
Method 1 – Listing Multiples
This is the most direct method — no prime knowledge needed, just counting. It’s the natural starting point for small numbers, but checking multiple after multiple by hand quickly becomes tedious once the numbers grow, which is exactly why Methods 2 and 3 exist.
Step by Step
- List the multiples of the largest number first, in order: 1×, 2×, 3×…
- For each multiple, check whether it’s also divisible by every other number in the problem.
- The first multiple that passes every check is the LCM.
8 is the larger number, so list its multiples and check each one against 6:
| Multiple of 8 | Divisible by 6? |
|---|---|
| 8 | No (8÷6 = 1.33) |
| 16 | No (16÷6 = 2.67) |
| 24 | Yes — 24÷6 = 4 ✓ |
9 is the largest, so list its multiples and check each one against both 4 and 6:
| Multiple of 9 | Divisible by 4? | Divisible by 6? |
|---|---|---|
| 9 | No | No |
| 18 | No (18÷4 = 4.5) | Yes |
| 27 | No | No |
| 36 | Yes — 36÷4 = 9 ✓ | Yes — 36÷6 = 6 ✓ |
Notice that 18 was divisible by 6 but not by 4 — a multiple has to satisfy every number in the list before it counts, not just some of them.
Method 2 – Prime Factorization
Instead of testing multiples one at a time, this method breaks every number down into its prime building blocks and combines them directly. It scales far better than listing multiples, especially with three or more numbers.
Step by Step
- Find the prime factorization of every number involved.
- Write each factorization in exponent form (e.g. 2² × 3¹).
- List every prime that appears in any of the numbers — not just the shared ones.
- For each prime, take whichever exponent is highest across all the numbers.
- Multiply those highest powers together — the result is the LCM.
14 = 2¹ × 7¹ 21 = 3¹ × 7¹
| Prime | In 14 | In 21 | Highest power used |
|---|---|---|---|
| 2 | 2¹ | — | 2¹ |
| 3 | — | 3¹ | 3¹ |
| 7 | 7¹ | 7¹ | 7¹ |
Unlike the GCF, every prime gets included here — even 2 and 3, which only appear in one of the two numbers — because the LCM needs to cover everything both numbers require.
12 = 2² × 3 18 = 2 × 3² 20 = 2² × 5
Highest powers across all three: 2², 3², 5¹
Method 3 – Using GCF
Once you already know how to find the GCF, this shortcut gives you the LCM of two numbers almost for free — no factoring, no listing, just one multiplication and one division.
Step by Step
- Find the GCF of the two numbers, using whichever GCF method you like (listing, prime factorization, or the ladder method).
- Multiply the two original numbers together: a × b.
- Divide that product by the GCF you found. The result is the LCM.
GCF(14, 21): factors of 14 = {1, 2, 7, 14}; factors of 21 = {1, 3, 7, 21}. Common factors: 1, 7 — GCF = 7.
LCM = (14 × 21) ÷ 7 = 294 ÷ 7 = 42
This matches the prime factorization example above exactly — both methods always agree.
Listing multiples of 180 by hand until one divides 48 evenly would take a while. The GCF formula gets there in three quick steps instead.
Step 1 — find GCF(48, 180) using the ladder method:
| Divide by | 48 | 180 |
|---|---|---|
| 2 | 24 | 90 |
| 2 | 12 | 45 |
| 3 | 4 | 15 |
4 and 15 share no common factor — GCF = 2 × 2 × 3 = 12
Step 2 — multiply the two numbers: 48 × 180 = 8,640.
Step 3 — divide by the GCF: 8,640 ÷ 12 = 720.
Method Comparison
| Method | Best For | Limitation |
|---|---|---|
| Listing multiples | Small numbers, quick checks | Slow for large or three-way LCM |
| Prime factorization | Three or more numbers | Requires factorization first |
| GCF formula | Exactly two numbers | Does not extend to three+ directly |
Key Takeaways
- Prime factorization: take the highest power of every prime that appears.
- GCF formula is the fastest for two numbers when GCF is easy to find.
- For three numbers: find LCM of the first two, then LCM of that result with the third.