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Er. Bhupendra
Er. Bhupendra

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LCM-HCF

LCM = Least Common Multiple
Matlab: sabse chhoti number jo diye gaye sab numbers se completely divide ho jaye.

Samajhne ka easiest real-life way:

Example 1: Traffic Signal 🚦

3 traffic lights hain:

  • Light A हर 2 min me green hoti hai
  • Light B हर 3 min me green hoti hai
  • Light C हर 4 min me green hoti hai

Question: Teeno ek saath kab green hongi?

Iske liye LCM निकालते हैं:

  • 2 → 2, 4, 6, 8, 12
  • 3 → 3, 6, 9, 12
  • 4 → 4, 8, 12

Sabme common sabse chhota = 12

➡️ Matlab 12 minutes baad teeno ek saath green.

Example 2: Medicines 💊

  • Medicine A every 6 hours
  • Medicine B every 8 hours
  • Medicine C every 12 hours

Kab ek saath leni padegi?

LCM(6, 8, 12)

Prime factor:

  • 6 = 2 × 3
  • 8 = 2³
  • 12 = 2² × 3

Highest power lo:

  • 2³ × 3 = 24

➡️ 24 hours baad teeno same time.

“LCM nikalne se hota kya?”

Ye mainly sync / repeat timing ke liye hota hai:

  • Events kab ek saath repeat honge
  • Machines kab same cycle pe aayengi
  • Bells / alarms कब साथ बजेंगे
  • Scheduling problems

2 ya 3 numbers me socho:

LCM of 2,3,5 ka matlab:

Aisa smallest number jo 2, 3, aur 5 sabse divide ho.

Example:
LCM(2,3,5) = 30
Kyuki:

  • 30 ÷ 2 ✔️
  • 30 ÷ 3 ✔️
  • 30 ÷ 5 ✔️

Memory trick 🧠:
LCM = “Sab kab milenge?”
HCF = “Sabme common kya hai?”

Bas itna yaad rakho:

  • LCM → Future meeting point

* HCF → Common shared part

HCF (Highest Common Factor) = sabse bada number jo sab numbers ko equally divide kar de.

Real life me HCF ka use tab hota hai jab kisi cheez ko equal parts me baantna ho without remainder.

Example 1: Chocolates 🍫

Tumhare paas:

  • 12 chocolates
  • 18 candies

Tum packets banana chahte ho jisme har packet me same number of chocolates aur candies ho.

Question: Maximum kitne packets bana sakte ho?

HCF(12,18)

Factors:

  • 12 → 1,2,3,4,6,12
  • 18 → 1,2,3,6,9,18

Sabse bada common = 6

➡️ 6 packets bana sakte ho
Har packet:

  • 12/6 = 2 chocolates
  • 18/6 = 3 candies

Example 2: Tiles / Cutting 📏

Ek room ka size:

  • Length = 24 m
  • Width = 36 m

Tum largest square tiles lagana chahte ho bina cut kiye.

Tile side = HCF(24,36)

HCF = 12

➡️ Largest square tile = 12m × 12m

Example 3: Rope Cutting 🪢

  • Rope A = 20 m
  • Rope B = 30 m

Equal longest pieces me cut karna hai.

HCF(20,30)=10

➡️ Dono ko 10m ke pieces me kaat sakte ho.

Simple intuition:

LCM vs HCF

  • LCM → “Sab kab milenge?” (time / cycles)
  • HCF → “Sabko equal max parts me kaise baantenge?”

Memory trick 🧠
HCF = Biggest common divider (baantna)

LCM = Smallest common multiple (milna)

Haan, short memory trick ke liye aise yaad kar sakte ho:

  • LCM → Multiple “Kaun sa number multiples me common aata hai?”

Example: 4 ke multiples → 4, 8, 12, 16...
6 ke multiples → 6, 12, 18...
Common multiple = 12 → LCM

  • HCF → Denominator / Divisor / Factor “Kaun sa number sabko divide (denominator) kar sakta hai?”

Example: 12 aur 18

  • 12 ÷ 6 = 2
  • 18 ÷ 6 = 3

6 dono ka common divisor/factor hai → HCF

Ek aur easy line:

🧠 LCM = Multiply side (upar badho)
🧠 HCF = Divide side (niche todo)

Ya:

  • LCM → Milna
  • HCF → Baantna

Isliye tumhara sentence almost sahi hai:
“Ek multiple hai, ek denominator/divisor hai.” ✔️

===============================================
Tumhari observation sahi hai — LCM aur HCF ek dusre ke mirror/reverse jaise lagte hain.
Confusion normally yahi hota hai. Main tumhari language me samjhata hoon.

Maan lo numbers hain: 12, 18

HCF: “Sabko equally divide karne wala bada number”

Tum bol rahe ho:

“Bahot saara data hai, ek aisa data nikalna jo sabko equally divide kar de”

Ye HCF hai.

Check:

  • 12 % 6 = 0
  • 18 % 6 = 0

6 sabko divide kar raha. Aur maximum bhi hai.

So HCF me hum niche jaate hain (divide side).

Visual:

12 → divide by ?
18 → divide by ?
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Question: Kaunsa biggest divisor common hai?

LCM: “Sabka common multiple chhota number”

Tum bol rahe ho:

“Ek aisa number chahiye jo diye hue sab numbers ka multiple ho aur chhota ho”

Ye LCM hai.

Check:

  • 36 % 12 = 0
  • 36 % 18 = 0

36 dono ka multiple hai. Aur common multiples me smallest.

So LCM me hum upar jaate hain (multiply side).

Visual:

12 → 12,24,36,48...
18 → 18,36,54...
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Question: Kaunsa smallest common multiple hai?

Tumhara “reverse” wala point

Bilkul:

HCF LCM
Input numbers ke andar common dhundo Input numbers ke bahar common dhundo
Divide Multiply
Biggest Smallest

Isliye mirror jaisa:

  • HCF: “Kitna tod sakta hu?”
  • LCM: “Kitna badha ke mila sakta hu?”

Memory trick 🧠

HCF = andar ghuso (factors)
LCM = bahar niklo (multiples)
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Agar interview me 1 line me bolna ho:

**HCF finds largest common divisor.

LCM finds smallest common divisible number.** ✔️

Real-life question me sabse pehle keywords pakdo.
Bas 2 sawal pucho:

Q1: Kya question “BAANTNE / DIVIDE” ki baat kar raha?

Agar haan → HCF

Keywords:

  • equally divide
  • same size pieces
  • maximum length / largest size
  • distribute without remainder
  • grouping / packing

Examples:

  • 24m aur 36m rope ko equal longest pieces me kaatna
  • 12 chocolates + 18 candies ko equal packets me baantna
  • Room me largest square tile lagani

➡️ Rule: Todna / Baantna = HCF

Q2: Kya question “EK SAATH / REPEAT / CYCLE” ki baat kar raha?

Agar haan → LCM

Keywords:

  • together again
  • after how much time
  • repeat cycle
  • common schedule
  • simultaneous

Examples:

  • 3 bells every 4, 6, 8 min → ek saath kab?
  • Medicines every 6h, 8h → same time kab?
  • Traffic lights kab saath green?

➡️ Rule: Milna / Repeat = LCM

Fast decision tree 🧠 

Question padha?
   |
   ├─ Equal parts / maximum piece / divide ?
   |          → HCF
   |
   └─ Together / cycle / repeat / common time ?
              → LCM
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One-line memory:

  • HCF = Cut / Divide / Pack
  • LCM = Meet / Repeat / Sync

Agar question bhejoge, main bata dunga kaise 5 sec me identify karna.

Haan — ye kaafi achha mental model hai. 👍

Tum aise soch sakte ho:

LCM = Integration / Merge / Combine

“Alag-alag data/cycles ko ek common point par lana”

Jaise:

  • 3 systems different intervals pe run ho rahe
  • Ek common sync point chahiye

Yaani multiple side → grow / merge / integrate

Example:

2 → 2,4,6,8,10,12...
3 → 3,6,9,12...
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Dono expand ho rahe until meet at 12.

➡️ LCM = integration feeling ✔️

HCF = Divide / Break / Partition

“Data ko equal pieces me todna”

Jaise:

  • 12 items
  • 18 items

Question:
Kitne equal groups / pieces bana sakte?

Yaani andar common divisor dhundo.

➡️ HCF = breaking / partition feeling ✔️

Tumhari language me final memory trick:

LCM → Integrate / Merge / Meet / Expand
HCF → Divide / Break / Partition / Equal pieces
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Aur ek deep intuition:

Ye dono almost reverse operations hain:

  • LCM: chhote numbers ko bada common number banate
  • HCF: bade numbers ko chhota common factor me todte

Isliye tum jo “reverse lag raha” bol rahe the — exactly wahi concept hai.

Tumhari intuition sahi direction me hai.

Time & Work me LCM ko total work maana initially weird lagta hai — “LCM se work kaise aa gaya?”
Tumhara doubt bahut common hai.

Main intuition se samjhata hu.

Maan lo:

  • A alone work finish karta hai in 4 days
  • B alone work finish karta hai in 6 days

Question: total work kitna maan lein?

Normally unknown hai:

Total Work = ?
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Hum aisa number choose karte jo 4 aur 6 dono se divide ho jaye, taki per-day work clean integer aaye.

4 aur 6 ka LCM = 12

To maan liya:

Total Work = 12 units
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Ab dekho magic:

A ka 1 day work

A 4 din me 12 units karta

So per day:

12 / 4 = 3 units/day
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B ka 1 day work

B 6 din me 12 units karta

So:

12 / 6 = 2 units/day
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Ab dono integer me aa gaye:

  • A → 3 units/day
  • B → 2 units/day

Together:

3 + 2 = 5 units/day
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Total work = 12 units

Time:

12 / 5 = 2.4 days
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But WHY LCM?

Ye important part hai.

Hum LCM isliye choose karte kyunki:

“Aisa total work choose karo jo sab workers ke days se clean divide ho.”

Agar total work 100 units maan lo:

  • A → 100/4 = 25
  • B → 100/6 = 16.666...

Fraction aa gaya.

LCM lene se usually:

  • calculation easy
  • fractions avoid
  • integer rates milte

Deep intuition

Tum LCM ko yaha aise socho:

Worker A ka cycle = 4 days
Worker B ka cycle = 6 days

Ek common work unit chahiye jo dono ke cycle me fit ho.

Exactly jaise pehle bola tha:

  • LCM = integration / common meeting point

Yaha:

  • 4-day worker
  • 6-day worker

Dono ka common work scale = 12 units

Ek line:

Time & Work me LCM actual total work nahi hota — calculation easy karne ke liye assumed total work hota hai. ✔️

Ye point pakad lo — sab clear ho jayega.

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