Factors and Multiples • Topic 5 of 7

Least Common Multiple (LCM)

The least common multiple (LCM) of two numbers is the smallest number both divide into. From prime factorization, take each prime raised to its highest power across the numbers: for 12 = 2² × 3 and 18 = 2 × 3², the LCM is 2² × 3² = 36. The LCM answers “when do events line up again” problems — bells ringing, lights flashing, schedules repeating. A reliable shortcut is LCM(a, b) = (a × b) ÷ GCF(a, b). For more than two numbers, build the LCM prime by prime, always keeping the highest exponent seen.

✅ Solved examples

1. Find the LCM of 12 and 18.
Factor: 12 = 2² × 3, 18 = 2 × 3². Take the highest power of each prime: 2² × 3² = 4 × 9 = 36. So LCM = 36.
2. Two bells ring every 8 and 12 minutes. If they ring together now, after how many minutes do they next ring together?
They coincide at the LCM of 8 and 12. 8 = 2³, 12 = 2² × 3, highest powers 2³ × 3 = 24. They ring together again after 24 minutes.
3. Find the LCM of 6 and 8 using the GCF shortcut.
GCF(6, 8) = 2. LCM = (6 × 8) ÷ GCF = 48 ÷ 2 = 24.
4. Find the LCM of 4, 6 and 10.
Factor: 4 = 2², 6 = 2 × 3, 10 = 2 × 5. Highest powers: 2² × 3 × 5 = 4 × 15 = 60. LCM = 60.

✏️ Practice — try these, take hints as needed

1. Find the LCM of 9 and 15.
Factor each: 9 = 3², 15 = 3 × 5.
Take the highest power of every prime that appears.
3² × 5.
45.
2. Hot dogs come in packs of 10 and buns in packs of 8. What is the least number of each you must buy to have equal counts?
You need a common multiple of 10 and 8.
“Least number” means the LCM.
10 = 2 × 5, 8 = 2³ → 2³ × 5.
40 of each.
3. Find the LCM of 7 and 11.
Both are prime.
They share no factors, so GCF = 1.
LCM of coprime numbers is their product.
77.
4. Lights flash every 15, 30 and 45 seconds. When do all three flash together (after the start)?
Find LCM(15, 30, 45).
15 = 3 × 5, 30 = 2 × 3 × 5, 45 = 3² × 5.
Highest powers: 2 × 3² × 5.
90 seconds.
5. The LCM of two numbers is 60 and their GCF is 5. If one number is 15, find the other.
Use GCF × LCM = product of the numbers.
5 × 60 = 300 = product.
Other = 300 ÷ 15.
20.

📝 Topic test — 8 questions

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