📌 Series Info: Ye post Chapter 1: Number System ka part hai.
Post Number: 4/76 | Topic: 1.4 HCF & LCM
Previous: 1.1 Classification ✅ | 1.2 Divisibility ✅ | 1.3 Prime Factorization ✅
Post Number: 4/76 | Topic: 1.4 HCF & LCM
Previous: 1.1 Classification ✅ | 1.2 Divisibility ✅ | 1.3 Prime Factorization ✅
Introduction: HCF Aur LCM Kya Hain?
Namaste dosto! Aaj ham HCF (Highest Common Factor) aur LCM (Least Common Multiple) padhenge. Ye SSC CGL ka sabse important aur high-weightage topic hai.
HCF Kya Hai?
🔹 HCF = Highest Common Factor (सबसे बड़ा उà¤à¤¯à¤¨िष्ठगुणनखंड)
Other names: GCD (Greatest Common Divisor), GCF (Greatest Common Factor)
Other names: GCD (Greatest Common Divisor), GCF (Greatest Common Factor)
Simple words me: Do ya zyada numbers ka sabse bada common factor (divisor) jo un sabko divide kar sake.
Example: 12 aur 18 ka HCF nikalo
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
Common factors: 1, 2, 3, 6
Highest common factor: 6 ✓
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
Common factors: 1, 2, 3, 6
Highest common factor: 6 ✓
LCM Kya Hai?
🔹 LCM = Least Common Multiple (न्यूनतम समापवर्त्य)
Sabse chhota common multiple
Sabse chhota common multiple
Simple words me: Do ya zyada numbers ka sabse chhota common multiple jo un sabse divisible ho.
Example: 4 aur 6 ka LCM nikalo
Multiples of 4: 4, 8, 12, 16, 20, 24...
Multiples of 6: 6, 12, 18, 24, 30...
Common multiples: 12, 24, 36...
Least common multiple: 12 ✓
Multiples of 4: 4, 8, 12, 16, 20, 24...
Multiples of 6: 6, 12, 18, 24, 30...
Common multiples: 12, 24, 36...
Least common multiple: 12 ✓
🎯 Exam Importance: SSC CGL me 2-3 direct questions aate hain. Plus Time & Work, Speed & Distance, aur bahut saare topics me indirectly use hota hai.
HCF Nikalne Ke Methods
HCF nikalne ke mainly 3 methods hain:
- Prime Factorization Method
- Division Method (Euclid's Algorithm) – Fastest ⚡
- Factor Listing Method – Basic
Method 1: Prime Factorization Method (HCF)
Steps:
- Dono numbers ka prime factorization karo
- Common prime factors nikalo
- Har common factor ki minimum power lo
- Un sabko multiply karo
Example 1: 24 aur 36 ka HCF nikalo
Step 1: Prime factorization
24 = 2³ × 3¹
36 = 2² × 3²
Step 2: Common factors identify karo
Common: 2 aur 3 ✓
Step 3: Minimum powers lo
2 ki power: min(3,2) = 2 → 2²
3 ki power: min(1,2) = 1 → 3¹
Step 4: Multiply karo
HCF = 2² × 3¹ = 4 × 3 = 12 ✓
Step 1: Prime factorization
24 = 2³ × 3¹
36 = 2² × 3²
Step 2: Common factors identify karo
Common: 2 aur 3 ✓
Step 3: Minimum powers lo
2 ki power: min(3,2) = 2 → 2²
3 ki power: min(1,2) = 1 → 3¹
Step 4: Multiply karo
HCF = 2² × 3¹ = 4 × 3 = 12 ✓
Example 2: 60, 90 aur 120 ka HCF nikalo
Prime factorization:
60 = 2² × 3 × 5
90 = 2 × 3² × 5
120 = 2³ × 3 × 5
Common factors: 2, 3, 5
Minimum powers:
2: min(2,1,3) = 1 → 2¹
3: min(1,2,1) = 1 → 3¹
5: min(1,1,1) = 1 → 5¹
HCF = 2 × 3 × 5 = 30 ✓
Prime factorization:
60 = 2² × 3 × 5
90 = 2 × 3² × 5
120 = 2³ × 3 × 5
Common factors: 2, 3, 5
Minimum powers:
2: min(2,1,3) = 1 → 2¹
3: min(1,2,1) = 1 → 3¹
5: min(1,1,1) = 1 → 5¹
HCF = 2 × 3 × 5 = 30 ✓
Method 2: Division Method (Euclid's Algorithm) FASTEST
Steps (2 Numbers Ke Liye):
- Bade number ko chhote number se divide karo
- Remainder nikalo
- Ab divisor ko dividend banao aur remainder ko divisor banao
- Jab remainder 0 ho jaye, us waqt ka divisor hi HCF hai
Example 1: 48 aur 18 ka HCF nikalo
Step 1: 48 ÷ 18 = Quotient 2, Remainder 12
Step 2: 18 ÷ 12 = Quotient 1, Remainder 6
Step 3: 12 ÷ 6 = Quotient 2, Remainder 0 ✓
Remainder 0 ho gaya, toh HCF = 6 ✓
Step 2: 18 ÷ 12 = Quotient 1, Remainder 6
Step 3: 12 ÷ 6 = Quotient 2, Remainder 0 ✓
Remainder 0 ho gaya, toh HCF = 6 ✓
Example 2: 1024 aur 256 ka HCF (bade numbers)
Step 1: 1024 ÷ 256 = Q=4, R=0 ✓
Pehle hi remainder 0, toh HCF = 256 ✓
Note: Jab ek number dusre se completely divide ho jaye, toh chhota number hi HCF hota hai.
Pehle hi remainder 0, toh HCF = 256 ✓
💡 Pro Tip: Division method sabse fast hai especially jab:
- Numbers bade hon (3-4 digits)
- Prime factorization difficult ho
- Exam me time bachana ho
3 Ya Zyada Numbers Ka HCF:
Example: 12, 18, 24 ka HCF nikalo
Method: Pehle do ka nikalo, phir result ko teesre ke saath divide karo
Step 1: HCF(12, 18)
18 ÷ 12 = R = 6
12 ÷ 6 = R = 0
HCF = 6
Step 2: HCF(6, 24)
24 ÷ 6 = R = 0
HCF = 6
Final HCF = 6 ✓
Method: Pehle do ka nikalo, phir result ko teesre ke saath divide karo
Step 1: HCF(12, 18)
18 ÷ 12 = R = 6
12 ÷ 6 = R = 0
HCF = 6
Step 2: HCF(6, 24)
24 ÷ 6 = R = 0
HCF = 6
Final HCF = 6 ✓
LCM Nikalne Ke Methods
LCM nikalne ke mainly 3 methods hain:
- Prime Factorization Method
- Common Division Method – Most Popular ⚡
- Formula Method (using HCF)
Method 1: Prime Factorization Method (LCM)
Steps:
- Sabhi numbers ka prime factorization karo
- Sabhi prime factors (common + uncommon) lo
- Har factor ki maximum power lo
- Sabko multiply karo
Example 1: 12 aur 18 ka LCM nikalo
Prime factorization:
12 = 2² × 3¹
18 = 2¹ × 3²
All prime factors: 2 aur 3
Maximum powers:
2: max(2,1) = 2 → 2²
3: max(1,2) = 2 → 3²
LCM = 2² × 3² = 4 × 9 = 36 ✓
Prime factorization:
12 = 2² × 3¹
18 = 2¹ × 3²
All prime factors: 2 aur 3
Maximum powers:
2: max(2,1) = 2 → 2²
3: max(1,2) = 2 → 3²
LCM = 2² × 3² = 4 × 9 = 36 ✓
Example 2: 15, 20, 25 ka LCM nikalo
Prime factorization:
15 = 3 × 5
20 = 2² × 5
25 = 5²
All factors: 2, 3, 5
Maximum powers:
2: max(0,2,0) = 2 → 2²
3: max(1,0,0) = 1 → 3¹
5: max(1,1,2) = 2 → 5²
LCM = 2² × 3 × 5² = 4 × 3 × 25 = 300 ✓
Prime factorization:
15 = 3 × 5
20 = 2² × 5
25 = 5²
All factors: 2, 3, 5
Maximum powers:
2: max(0,2,0) = 2 → 2²
3: max(1,0,0) = 1 → 3¹
5: max(1,1,2) = 2 → 5²
LCM = 2² × 3 × 5² = 4 × 3 × 25 = 300 ✓
Method 2: Common Division Method MOST POPULAR
Steps:
- Sabhi numbers ko ek line me likho
- Sabse chhote prime number se divide karo (jitne divide ho sakein)
- Jo divide na hon unhe waise hi niche likho
- Jab tak sab numbers 1 na ho jayein, repeat karo
- Left side ke sabhi divisors ko multiply karo
Example 1: 12, 18, 24 ka LCM
2 | 12, 18, 24
2 | 6, 9, 12
2 | 3, 9, 6
3 | 3, 9, 3
3 | 1, 3, 1
| 1, 1, 1
LCM = 2 × 2 × 2 × 3 × 3 = 2³ × 3² = 72 ✓
2 | 6, 9, 12
2 | 3, 9, 6
3 | 3, 9, 3
3 | 1, 3, 1
| 1, 1, 1
LCM = 2 × 2 × 2 × 3 × 3 = 2³ × 3² = 72 ✓
Example 2: 8, 12, 16 ka LCM
2 | 8, 12, 16
2 | 4, 6, 8
2 | 2, 3, 4
2 | 1, 3, 2
3 | 1, 3, 1
| 1, 1, 1
LCM = 2 × 2 × 2 × 2 × 3 = 2⁴ × 3 = 48 ✓
2 | 4, 6, 8
2 | 2, 3, 4
2 | 1, 3, 2
3 | 1, 3, 1
| 1, 1, 1
LCM = 2 × 2 × 2 × 2 × 3 = 2⁴ × 3 = 48 ✓
💡 Exam Tip: Common division method sabse fast aur error-free hai. Multiple numbers ka LCM ek saath nikal sakte ho!
HCF & LCM Ka Golden Formula MUST KNOW
⭐ HCF × LCM = First Number × Second Number ⭐
Ye formula sirf 2 numbers ke liye work karta hai!
Example: 12 aur 18 ka HCF = 6, LCM = 36
Verify: HCF × LCM = 6 × 36 = 216
12 × 18 = 216 ✓
Match ho gaya! Formula correct ✓
Verify: HCF × LCM = 6 × 36 = 216
12 × 18 = 216 ✓
Match ho gaya! Formula correct ✓
Formula Ka Use:
Question: Do numbers ka HCF 12 hai aur unka product 1728 hai. LCM kya hoga?
Solution:
HCF × LCM = Product of numbers
12 × LCM = 1728
LCM = 1728 ÷ 12
LCM = 144 ✓
Solution:
HCF × LCM = Product of numbers
12 × LCM = 1728
LCM = 1728 ÷ 12
LCM = 144 ✓
HCF vs LCM – Key Differences
| Feature | HCF | LCM |
|---|---|---|
| Full Form | Highest Common Factor | Least Common Multiple |
| Meaning | Sabse bada common divisor | Sabse chhota common multiple |
| Value | Hamesha numbers se chhota ya equal | Hamesha numbers se bada ya equal |
| Prime Method | Common factors ki minimum powers | All factors ki maximum powers |
| Example | HCF(12,18) = 6 | LCM(12,18) = 36 |
| Use Case | Division problems, grouping | Timing problems, repetition |
Special Cases & Shortcuts
Case 1: Co-Prime Numbers (Sahaapeksh Sankhya)
Agar do numbers ka HCF = 1, toh wo co-prime kehlate hain.
Co-prime numbers ka LCM = unka product
Co-prime numbers ka LCM = unka product
Example: 7 aur 11 (dono prime hain)
HCF(7,11) = 1 (co-prime) ✓
LCM(7,11) = 7 × 11 = 77 ✓
HCF(7,11) = 1 (co-prime) ✓
LCM(7,11) = 7 × 11 = 77 ✓
Case 2: One Number Divides Another
Agar ek number dusre number ko completely divide karta hai:
HCF = Chhota number
LCM = Bada number
HCF = Chhota number
LCM = Bada number
Example: 12 aur 36
36 ÷ 12 = 3 (completely divisible)
HCF(12,36) = 12 ✓
LCM(12,36) = 36 ✓
36 ÷ 12 = 3 (completely divisible)
HCF(12,36) = 12 ✓
LCM(12,36) = 36 ✓
Case 3: Prime Numbers
Do prime numbers ka:
HCF = 1 (always)
LCM = unka product
HCF = 1 (always)
LCM = unka product
Case 4: Consecutive Numbers
Do consecutive numbers (like 5,6 or 11,12) ka:
HCF = 1 (always)
LCM = unka product
HCF = 1 (always)
LCM = unka product
Example: 15 aur 16
HCF = 1 ✓
LCM = 15 × 16 = 240 ✓
HCF = 1 ✓
LCM = 15 × 16 = 240 ✓
Real Life Applications (Word Problems)
HCF Type Questions:
Q1. Ek rectangle ki length 12 cm aur breadth 18 cm hai. Isse sabse badi square tiles se cover karna hai (without cutting). Har tile ki side