LCM–HCF Mixed Applications

🎯 Learning Objective

Solve product-relation questions and word problems combining both concepts.

💡 Concept

• For two numbers: a × b = HCF × LCM
• If HCF = h, numbers can be written as hx and hy where x, y are co-prime
• LCM is always a multiple of HCF
• If HCF and LCM are given and one number known → other = (HCF × LCM) ÷ known

🧮 Key Formulas

a × b = HCF × LCM

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Numbers = h·x, h·y with x,y co-prime

✏️ Easy Example

Q. HCF of two numbers is 4, LCM is 48. If one number is 12, find the other.

1. Other = (HCF × LCM) ÷ 12
2. = (4 × 48) ÷ 12 = 192 ÷ 12

Answer: 16

🇮🇳 Real-Life Example

Two gear wheels with 12 and 16 teeth mesh together; the product rule links how often teeth realign (LCM) with the shared spacing (HCF).

📝 Exam-Level Example

Q. The sum of two numbers is 36 and their HCF is 4. How many such pairs are possible?

1. Numbers = 4x and 4y → 4(x+y) = 36 → x+y = 9
2. Co-prime pairs summing to 9: (1,8), (2,7), (4,5)
3. Count = 3

Answer: 3 pairs

🪄 Memory Trick

Ratio given? Numbers in ratio a:b with HCF h → numbers are ha and hb, LCM = h·a·b.

⚠️ Common Mistakes

• ❌ Applying a×b = HCF×LCM to three numbers
• ❌ Choosing non-co-prime x, y in pair questions

🏆 Exam Tips

• ✅ Check divisibility: LCM ÷ HCF must be a whole number
• ✅ Ratio + HCF → LCM = HCF × product of ratio terms

📌 Summary

• a×b = HCF × LCM (two numbers)
• Numbers = h·x, h·y (x,y co-prime)
• Ratio a:b, HCF h → LCM = h·a·b