Divisibility, Factors & Multiples • Topic 4 of 5

LCM

The least common multiple LCM(a, b) is the smallest positive integer that both a and b divide into. By prime factorisation, take each prime that appears in either number raised to its highest power. For 12 = 2²·3 and 18 = 2·3², use 2² and 3², giving LCM = 4 × 9 = 36. LCM answers "when do repeating cycles line up?" — two buses that leave every 12 and 18 minutes next depart together after 36 minutes; it is also the right common denominator when adding fractions. Contrast it with GCD: GCD is at most the smaller number, LCM is at least the larger. The GRE frequently disguises LCM as a scheduling or "meet again" word problem, so when two independent cycles must coincide, reach for LCM.

✅ Solved examples

1. Find LCM(12, 18).
12 = 2²·3, 18 = 2·3². Take the highest power of each prime: 2² and 3². LCM = 4 × 9 = 36.
2. Two lighthouses flash every 8 and 12 seconds and just flashed together. When do they next flash together?
They coincide at LCM(8, 12). 8 = 2³, 12 = 2²·3, so LCM = 2³·3 = 24. Next simultaneous flash: 24 seconds later.
3. What is the smallest number divisible by 4, 5 and 6?
LCM(4, 5, 6): 4 = 2², 5 = 5, 6 = 2·3. Highest powers: 2²·3·5 = 60.
4. Quantitative Comparison. Quantity A: LCM(6, 8). Quantity B: 6 × 8. Which is greater?
LCM(6, 8) = 24 (since 6 = 2·3, 8 = 2³ → 2³·3). 6 × 8 = 48. Quantity B is greater — the product overshoots because 6 and 8 share the factor 2.

✏️ Practice — try these, take hints as needed

1. Find LCM(10, 15).
10 = 2·5, 15 = 3·5.
Highest powers.
2·3·5.
30
2. Find LCM(9, 12).
9 = 3², 12 = 2²·3.
Take 3² and 2².
9 × 4.
36
3. Smallest number divisible by 3, 4 and 10?
LCM of all three.
Primes 2², 3, 5.
Multiply.
60
4. Trains leave every 15 and 25 minutes together at 9:00. Next joint departure?
LCM(15, 25).
15 = 3·5, 25 = 5².
3·5² = 75 min.
10:15
5. Find LCM(7, 11).
Both prime.
Coprime numbers.
LCM = product.
77

📝 Topic test — 8 questions

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