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 1 – Listing Multiples
List multiples of the largest number until you find one divisible by the others.
LCM(6, 8) by listing
Multiples of 8: 8, 16, 24, 32...
Is 8 divisible by 6? No. Is 16? No. Is 24? Yes (24÷6=4). LCM = 24
Method 2 – Prime Factorization
Write each number as a product of primes. Take each prime to its highest power across all the numbers. Multiply.
LCM(12, 18, 20)
12 = 2² × 3 18 = 2 × 3² 20 = 2² × 5
Highest powers: 2², 3², 5¹
LCM = 4 × 9 × 5 = 180
Method 3 – Using GCF
Apply the formula: LCM(a,b) = (a × b) ÷ GCF(a,b)
LCM(14, 21)
GCF(14,21): factors of 14 = {1,2,7,14}; factors of 21 = {1,3,7,21}. GCF = 7.
LCM = (14 × 21) ÷ 7 = 294 ÷ 7 = 42
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.
Practice Questions
- Find LCM(15, 20) using the GCF formula.
- Find LCM(8, 12, 18) using prime factorization.
- Find LCM(35, 50) using any method.
- Verify: LCM(9,12) × GCF(9,12) = 9 × 12.
- Find LCM(6, 10, 15) using prime factorization.