HCF — Highest Common Factor

🎯 Learning Objective

Find the HCF of numbers by factorisation and division method, and know when a question is asking for HCF.

💡 Concept

• HCF = the LARGEST number that divides all given numbers exactly
• Method 1 (prime factors): take COMMON primes with LOWEST powers
• Method 2 (division): divide bigger by smaller; keep dividing divisor by remainder till remainder 0
• Keywords that mean HCF: 'greatest', 'largest', 'maximum size', 'divides exactly', 'equal groups'

🧮 Key Formulas

HCF × LCM = Product of two numbers

✏️ Easy Example

Q. Find the HCF of 24 and 36.

1. 24 = 2³ × 3
2. 36 = 2² × 3²
3. Common with lowest powers: 2² × 3 = 12

Answer: 12

🇮🇳 Real-Life Example

Cutting two ribbons of 24 m and 36 m into equal largest pieces with nothing wasted → each piece 12 m. That's HCF.

📝 Exam-Level Example

Q. The largest number dividing 62, 132 and 237 leaving the same remainder in each case?

1. Same remainder → take differences: 132−62 = 70, 237−132 = 105
2. Required = HCF(70, 105)
3. 70 = 2×5×7, 105 = 3×5×7 → HCF = 35

Answer: 35

🪄 Memory Trick

'Same remainder' → HCF of DIFFERENCES. 'Remainders given' → subtract remainders first, then HCF.

⚠️ Common Mistakes

• ❌ Taking highest powers instead of lowest for HCF
• ❌ Forgetting HCF ≤ smallest number

🏆 Exam Tips

• ✅ HCF of co-primes is always 1
• ✅ Verify: HCF must divide every given number

📌 Summary

• HCF = largest common divisor
• Common primes, lowest powers
• 'Greatest/largest/equal pieces' → HCF
• Same remainder → HCF of differences