LCM — Lowest Common Multiple

🎯 Learning Objective

Find LCM quickly and recognise LCM-type questions (bells, traffic lights, circular tracks).

💡 Concept

• LCM = the SMALLEST number divisible by all given numbers
• Prime factor method: take ALL primes with HIGHEST powers
• Keywords for LCM: 'together again', 'smallest number', 'minimum', 'ring together', 'meet again'
• LCM is always ≥ the largest given number

🧮 Key Formulas

LCM = product of all primes with highest powers

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HCF × LCM = a × b (two numbers only)

✏️ Easy Example

Q. Find the LCM of 12 and 18.

1. 12 = 2² × 3
2. 18 = 2 × 3²
3. All primes, highest powers: 2² × 3² = 36

Answer: 36

🇮🇳 Real-Life Example

Two Mumbai local trains leave every 12 min and 18 min. Both leave together now → next together after 36 minutes. LCM!

📝 Exam-Level Example

Q. Four bells toll at intervals of 4, 6, 8 and 12 seconds. They toll together now; after how long will they toll together again?

1. LCM(4,6,8,12)
2. 4=2², 6=2×3, 8=2³, 12=2²×3
3. LCM = 2³ × 3 = 24

Answer: 24 seconds

📝 Exam-Level Example

Q. Smallest number which when divided by 6, 8, 15 leaves remainder 2 in each case?

1. LCM(6,8,15) = 120
2. Add the common remainder: 120 + 2

Answer: 122

🪄 Memory Trick

'Leaves remainder r in each case' → LCM + r. 'Leaves remainder (divisor−k) in each' → LCM − k.

⚠️ Common Mistakes

• ❌ Mixing up: HCF uses lowest powers, LCM uses highest
• ❌ Using HCF×LCM = product for THREE numbers (works only for two)

🏆 Exam Tips

• ✅ Bells/lights/tracks → straight LCM
• ✅ For fractions: LCM = LCM(numerators)/HCF(denominators)

📌 Summary

• LCM = smallest common multiple
• All primes, highest powers
• 'Together again' → LCM
• LCM+r and LCM−k remainder patterns