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HCF / LCM

Lesson 38 of 113

1. Ye topic hai kya?

HCF (GCD) = sabse bada number jo dono (ya teeno) ko divide kare. LCM = sabse chhota number jisme dono baith jaayein (unke multiples ka pehla common). Paper: tile kitni badi, ghanti kab saath, tape kitni lambi, “divided by 8,12,15 rem 3”.

HCF socho jab

Greatest / largest / maximum jo kaate / measure kare / square tile. Rooms ki sabse lambi tape. Equal rows of soldiers. Same remainder chhod ke kaate.

LCM socho jab

Least / smallest / first time saath. Bells, lights, runners. Minimum length jo dono se kat-e. 4-digit smallest multiple. “Leaves remainder r” with LCM×k+r.

Do numbers: HCF × LCM = a × b (teen pe nahi — Real numbers pe kyun). Coprime (HCF=1) → LCM = product. Tool (Euclid, min/max prime) wahan; yahan kaun sa formula, kaun sa word.
Divisibility “258 ko 3 kat-ta” Number system. 12,18 ka HCF=6 sirf tool-demo Real numbers pe ho chuka — yahan 48 aur 180, bells 16-20-24. Number series agla term nahi. Surface areas ki tile paint nahi — yahan floor cover.

2. Prime se — min HCF, max LCM

Har number toot: 48 = 24×3, 180 = 22×32×5. HCF = common primes ki chhoti power. LCM = har prime ki badi power (jo kisi mein ho).

Example

48 aur 180. HCF, LCM? Check product se.

Solution

  1. HCF min

    22×3 = 12. (24 nahi — 180 mein sirf 22.)

  2. LCM max

    24×32×5 = 16×9×5 = 720.

  3. Check

    12×720 = 8640, 48×180 = 8640. Teen numbers pe ye check mat. 12,18,36 wala Real numbers — copy mat.

AnswerHCF 12, LCM 720
Example

HCF=8, LCM=48, ek number 16. Doosra?

Solution

  1. Do-number

    Doosra = (8×48)/16 = 384/16 = 24. 16=24, 24=23×3, HCF=8, LCM=48. Diya hua HCF ka multiple hona chahiye — 16, 8 se kat-ta.

Answer24
Fractions: HCF = HCF(numerators) / LCM(denominators). LCM = LCM(nums) / HCF(dens). 3/8 aur 9/20: HCF = 3/40, LCM = 9/4. (3/8, 9/20 dono 3/40 ke multiples; 9/4 dono ka multiple.)

Teen: pehle do, phir teesra. 16, 20, 24 ka LCM agle section mein. Euclid lambi theory Real numbers; yahan listing/primes kaafi.

3. Bells, tiles, tape, saath

Ek line: kaatna / fit / measure → HCF. Cover as multiple / repeat until same moment → LCM.

0 16 20 24 240 s saath

Teen bells 16 s, 20 s, 24 s. Pehli common tick = LCM = 240 s = 4 min. HCF=4 sirf “common beat size”, saath aane ka time nahi.

Example

Bells 16, 20, 24 seconds. 7:00 pe saath. Agli baar kab? (Number system Q10 yahi tha — ab yahan.)

Solution

  1. Primes

    16=24, 20=22×5, 24=23×3. LCM=24×3×5=240 s = 4 min.

  2. Clock

    7:04. HCF=4 se 7:00:04 mat. 12,18,24 ka LCM 72 alag pair.

Answer7:04
16 12

Floor 16 × 12. Sabse badi square tile = HCF(16,12)=4. Kitni tiles = (16/4)×(12/4)=4×3=12. LCM=48 yahan galat (woh ribbon jo dono lengths ka multiple ho).

Example

Hall 8.4 m × 6.3 m. Sabse badi square tile (cm)? Kitni tiles?

Solution

  1. cm

    840 aur 630. Euclid: 840=630×1+210, 630=210×3+0. HCF=210 cm = 2.1 m.

  2. Count

    (840/210)×(630/210)=4×3=12. Area ratio (8.4×6.3)/(2.1)2 same 12. 4πr² sphere nahi.

Answer2.1 m ; 12 tiles
Example

Teen rooms 7.7 m, 5.5 m, 3.3 m. Sabse lambi tape jo teeno ko exact measure kare?

Solution

  1. cm

    770, 550, 330. 770=2×5×7×11, 550=2×5²×11, 330=2×3×5×11. HCF=2×5×11=110 cm = 1.1 m. LCM lambi tape nahi — LCM 11.55 m room se badi ho sakti, “exact measure” = kaatna = HCF.

Answer1.1 m
Example

Lights 48, 72, 96 seconds pe blink. Saath kab (seconds)?

Solution

  1. LCM

    48=24×3, 72=23×32, 96=25×3. Max=25×32=32×9=288 s.

Answer288 s
Soldiers 105, 140, 175 ko equal rows, sabse badi row: HCF=35 (7 common, 5 common). Har camp 3, 4, 5 rows. “Average 40 soldiers” Average chapter.

4. Remainder patterns

Do classic. (A) same remainder r jab a,b,c se divide: number = LCM(a,b,c)×k + r. (B) remainder har baar divisor se 1 kam: = LCM×k − 1. (C) greatest number jo x,y ko divide karke same rem r chhode: HCF(x−r, y−r) ya HCF of differences.

Least +ve: k=0 tab r khud, lekin r < har divisor hona chahiye; warna k=1. “Leaves remainder 3 with 8,12,15” → 120k+3, chhota 123 (3, 8 se chhota, 8 se divide nahi hota as “leaving 3” in the sense of 3=3 rem when n=3, but 3÷8 rem 3 yes actually… 3÷8 = 0 rem 3. Some papers 123 maangte as > divisors). LDC aksar 123.
Example

Sabse chhota number jo 8, 12, 15 se divide pe remainder 3 chhode?

Solution

  1. LCM

    8=23, 12=22×3, 15=3×5 → 120.

  2. +3

    120k+3. k=1 → 123. Check: 123=15×8+3, 12×10+3, 8×15+3. 3 khud bhi rem 3 deta lekin paper 123. Binary 101101 nahi.

Answer123
Example

5 se rem 4, 6 se rem 5, 7 se rem 6. Sabse chhota?

Solution

  1. Har 1 kam

    N+1, 5,6,7 se kat-ta. N+1 = LCM(5,6,7)×k = 210k. N=210k−1. k=1 → 209.

Answer209
Example

Sabse bada number jo 70 aur 125 ko divide karke remainder 4 chhode?

Solution

  1. Minus r

    HCF(66, 121). 66=2×3×11, 121=112. HCF=11. 70=6×11+4, 125=11×11+4. (HCF of 70 aur 125 = 5, galat — rem hataana pehle.)

Answer11
Example

Sabse chhota 4-digit jo 16, 18, 24 se kat-ta? Sabse bada 3-digit?

Solution

  1. LCM

    16=24, 18=2×32, 24=23×3 → 24×32=144.

  2. 4-digit

    1000/144 ≈ 6.94 → 7×144=1008.

  3. 3-digit max

    999/144 ≈ 6.94 → 6×144=864.

Answer1008 ; 864

5. Sawal — basic se pro

Q1 · Basic

48, 180. HCF?

  1. 12
  2. 6
  3. 48
  4. 720

Solution

  1. 2²×3

    12. D LCM.

Answer12
Q2 · Basic

48, 180. LCM?

  1. 720
  2. 12
  3. 8640
  4. 180

Solution

  1. 2⁴×3²×5

    720. C product.

Answer720
Q3 · Basic

HCF=8, LCM=48, ek=16. Doosra?

  1. 24
  2. 32
  3. 8
  4. 96

Solution

  1. 384/16

    24.

Answer24
Q4 · Basic

Bells 16, 20, 24 s. Saath after?

  1. 240 s
  2. 4 s
  3. 60 s
  4. 3840 s

Solution

  1. LCM 240

    B HCF. D product.

Answer240 s
Q5 · Medium

16×12 floor. Largest square tile side?

  1. 4
  2. 48
  3. 2
  4. 12

Solution

  1. HCF

    4. B LCM.

Answer4
Q6 · Medium

8.4 m × 6.3 m. Tile side?

  1. 2.1 m
  2. 21 m
  3. 1.05 m
  4. 8.4 m

Solution

  1. HCF 210 cm

    2.1 m.

Answer2.1 m
Q7 · Medium

8,12,15 se rem 3. Least?

  1. 123
  2. 120
  3. 3
  4. 117

Solution

  1. 120+3

    123. B LCM. D 120−3.

Answer123
Q8 · Medium

5 rem 4, 6 rem 5, 7 rem 6. Least?

  1. 209
  2. 210
  3. 211
  4. 29

Solution

  1. 210−1

    209.

Answer209
Q9 · Pro

70 aur 125, rem 4. Greatest divisor?

  1. 11
  2. 5
  3. 4
  4. 66

Solution

  1. HCF(66,121)

    11. B HCF(70,125).

Answer11
Q10 · Pro

(101101)2=45. Is chapter?

  1. Nahi — Number system
  2. Haan, HCF 45
  3. LCM binary
  4. Bells 45 s

Solution

  1. Binary

    A.

AnswerA

Practice

Pehle khud, phir neeche kholo.

12345 678910

P1

16×12 tiles. Kitni squares side 4?

  1. 12
  2. 4
  3. 48
  4. 192

Solution

  1. 4×3

    12. C LCM. D area.

Answer12
P2

8.4×6.3. Kitni tiles side 2.1 m?

  1. 12
  2. 4
  3. 2
  4. 210

Solution

  1. 4×3

    12.

Answer12
P3

Rooms 7.7, 5.5, 3.3 m. Longest tape?

  1. 1.1 m
  2. 11 m
  3. 3.3 m
  4. 16.5 m

Solution

  1. HCF 110 cm

    1.1 m. D LCM-ish.

Answer1.1 m
P4

Lights 48, 72, 96 s. Together?

  1. 288 s
  2. 24 s
  3. 96 s
  4. 216 s

Solution

  1. 2⁵×3²

    288. B HCF=24.

Answer288 s
P5

Soldiers 105,140,175. Largest equal row?

  1. 35
  2. 5
  3. 7
  4. 105

Solution

  1. HCF

    35.

Answer35
P6

Least 4-digit ÷ 16,18,24?

  1. 1008
  2. 144
  3. 1000
  4. 864

Solution

  1. 7×144

    1008. D 3-digit max.

Answer1008
P7

Greatest 3-digit ÷144?

  1. 864
  2. 1008
  3. 999
  4. 720

Solution

  1. 6×144

    864.

Answer864
P8

HCF 3/8 aur 9/20?

  1. 3/40
  2. 9/4
  3. 3/20
  4. 27/160

Solution

  1. 3/LCM(8,20)

    3/40. B LCM of fractions.

Answer3/40
P9

LCM 3/8 aur 9/20?

  1. 9/4
  2. 3/40
  3. 27/28
  4. 9/20

Solution

  1. 9/HCF(8,20)

    9/4.

Answer9/4
P10

√2 irrational proof. Is chapter?

  1. Nahi — Real numbers
  2. Haan, HCF(p,q)=1
  3. LCM of 2
  4. Tiles √2

Solution

  1. Proof

    A. Lowest-terms HCF tool wahan, bells yahan.

AnswerA

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Agle topic: Simplification / BODMAS / squares-cubes.