08. HCF & LCM (Methods + Properties)

HCF (GCD) aur LCM ka complete concept: methods (prime factorization, division, Euclid), properties, HCF×LCM relation, word-problems basics, SSC CGL level examples & practice with answers.

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1) HCF (GCD) kya hota hai?

HCF (Highest Common Factor) ya GCD (Greatest Common Divisor) wo sabse bada number hota hai jo given dono numbers ko completely divide kare.

Example: 12 aur 18 ka HCF = 6 (because 6 divides both)

2) LCM kya hota hai?

LCM (Least Common Multiple) wo sabse chhota positive number hota hai jo given dono numbers se divisible ho.

Example: 12 aur 18 ka LCM = 36

3) Best relation (SSC favourite)

Do numbers a aur b ke liye: HCF(a,b) × LCM(a,b) = a × b

• Example: a=12, b=18 → HCF=6, LCM=36 → 6×36 = 216 = 12×18 ✅

Note: Ye relation sirf 2 numbers ke liye direct use hota hai. 3 numbers me direct product relation generally use nahi hota.

4) Method 1: Prime Factorization (most common)

Steps: (1) Dono numbers ka prime factorization nikalo. (2) HCF ke liye common primes ki minimum power lo. (3) LCM ke liye saare primes ki maximum power lo.

Example: 72 aur 120

• 72 = 2³ × 3²
• 120 = 2³ × 3 × 5
• HCF = 2³ × 3¹ = 8×3 = 24
• LCM = 2³ × 3² × 5 = 8×9×5 = 360

5) Method 2: Division method (quick for 2–3 numbers)

Dono numbers ko common primes se divide karte jao. Divisors (jo divide kiye) ka product = HCF/LCM ke part ke roop me use hota. Practically, SSC me division method se LCM fast nikalta hai.

LCM example: 12, 18, 30

2 se divide: (12,18,30) → (6,9,15) 3 se divide: (6,9,15) → (2,3,5) 2 se divide: (2,3,5) → (1,3,5) 3 se divide: (1,3,5) → (1,1,5) 5 se divide: (1,1,5) → (1,1,1) LCM = 2×3×2×3×5 = 180

6) Method 3: Using HCF to find LCM (two numbers)

Agar HCF known hai, to: LCM = (a × b) / HCF

Example: a=48, b=60, HCF=12 ⇒ LCM = (48×60)/12 = 240

7) Properties (must know)

• HCF always ≤ smaller number
• LCM always ≥ bigger number
• If numbers are co-prime (HCF=1) ⇒ LCM = a×b
• If one number divides the other (a | b) ⇒ HCF = a, LCM = b (assuming a < b)

8) Small word-problem idea (concept)

“Bells/alarms together”, “traffic lights together”, “repeat cycle” type problems me mostly LCM use hota hai. “Max equal pieces”, “greatest length that divides”, “largest possible size” type problems me mostly HCF use hota hai.

• Example (LCM): 6 min aur 8 min interval — together again after LCM(6,8)=24 min
• Example (HCF): 24m aur 36m rope ko equal pieces — max piece length HCF(24,36)=12m

9) Common traps

• HCF vs LCM confusion in word problems (cycle=LCM, max equal pieces=HCF).
• Prime factorization me powers galat lena (HCF=min power, LCM=max power).
• 2 numbers ke product relation ko 3 numbers par apply mat karo.

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10) Practice (SSC CGL) + Answers

1. Find HCF and LCM of 18 and 24.
2. Find HCF and LCM of 72 and 120.
3. If HCF(45, 75)=15, then find LCM.
4. Find LCM of 8, 12, 15.
5. Two numbers are co-prime and their product is 221. Find their LCM.
6. Find HCF of 96 and 144.

Show Answers

1. 18=2×3², 24=2³×3 ⇒ HCF=2×3=6, LCM=2³×3²=72
2. HCF=24, LCM=360
3. LCM = (45×75)/15 = 225
4. 8=2³, 12=2²×3, 15=3×5 ⇒ LCM=2³×3×5=120
5. Co-prime ⇒ LCM = product = 221 (numbers 13 and 17)
6. 96=2⁵×3, 144=2⁴×3² ⇒ HCF=2⁴×3=48