Ch 2: HCF & LCM | Maths — HSSC CET Group D

Dosto, HCF (Highest Common Factor / Mahattam Samapvartak) aur LCM (Lowest Common Multiple / Laghuttam Samapvartya) — Maths ke sabse scoring topics mein se hain.

HSSC CET Group D mein isse 1–2 questions lagbhag har baar aate hain — zyada tar word problems: "teen ghantiyan kab saath bajengi?", "sabse bada number kaunsa jo teeno ko divide kare?" — inke fix patterns hain jo hum yahan seekhenge.

Pehle basic method, phir 4 famous question-types, phir tricks. Chapter 1 (divisibility rules) padh ke aaye ho to ye aur aasan lagega! 💪

1. Factor aur Multiple — pehle ye samjho

Factor (Gunankhand) = jo number kisi ko poora divide kare. 12 ke factors: 1, 2, 3, 4, 6, 12.
Multiple (Gunaj) = table wale numbers. 12 ke multiples: 12, 24, 36, 48, …
HCF = common factors mein sabse BADA. Ye numbers se chhota ya barabar hota hai.
LCM = common multiples mein sabse CHHOTA. Ye numbers se bada ya barabar hota hai.

📌 Ek line mein yaad karo HCF = "sabse bada divide karne wala" · LCM = "sabse chhota jo sabki table mein aaye"

2. Prime Factorisation Method

✏️ Solved Example — HCF aur LCM of 24, 36 Step 1: Prime factors likho: 24 = 2×2×2×3 = 2³×3 · 36 = 2×2×3×3 = 2²×3²
Step 2 (HCF): Common primes ki SABSE CHHOTI power: 2² × 3 = 12
Step 3 (LCM): Sab primes ki SABSE BADI power: 2³ × 3² = 8 × 9 = 72

Division method (LCM ka fast tarika): numbers ko saath likho, common primes se divide karte jao jab tak 1 na aa jaye — sab divisors ka product = LCM. Jaise 12, 18 → 2 se (6, 9) → 3 se (2, 3) → ab 2 aur 3 → LCM = 2×3×2×3 = 36.

3. MASTER FORMULA (sirf 2 numbers ke liye)

⭐ HCF × LCM = Pehla number × Doosra number Solved Example: HCF = 12, LCM = 72, ek number = 24. Doosra = ?
Step 1: 12 × 72 = 24 × doosra number
Step 2: Doosra = (12 × 72) ÷ 24 = 864 ÷ 24 = 36 ✓

4. Fractions ka HCF & LCM

Formula	Kaise
HCF of fractions	HCF (uparwale) ÷ LCM (nichewale)
LCM of fractions	LCM (uparwale) ÷ HCF (nichewale)

✏️ Solved Example — HCF of 2/3 aur 4/9 Step 1: Uparwale (2, 4) ka HCF = 2
Step 2: Nichewale (3, 9) ka LCM = 9
Step 3: Answer = 2/9 ✓ (Waise hi LCM of 2/5, 3/10 = LCM(2,3)/HCF(5,10) = 6/5)

5. Word Problems ke 4 FAMOUS Types

Question mein likha ho	Kya lagana hai
"Kab saath/ek saath…" (bells, lights, circular track)	LCM of intervals
"Sabse bada number jo … ko divide kare"	HCF
"Sabse chhota number jo … se divide ho"	LCM
"Divide kare aur remainder r bache"	Niche dekho ⬇️ (Type 2 & 3)

✏️ Type 1 — Bells problem (LCM) Teen bells 6, 8 aur 12 second par bajti hain. Ek saath bajne ke baad phir kab saath bajengi?
Step 1: LCM(6, 8, 12) nikalo: 6=2×3, 8=2³, 12=2²×3 → LCM = 2³×3 = 24 second ✓

✏️ Type 2 — "Sabse bada number, remainder same bache" (HCF of differences) Wo sabse bada number jo 62, 132 aur 237 ko divide kare aur har baar remainder 2 bache?
Step 1: Remainder MINUS karo: 62−2 = 60, 132−2 = 130, 237−2 = 235
Step 2: HCF(60, 130, 235) = 5 ✓ (check: 62÷5 → rem 2 ✓)

✏️ Type 3 — "Sabse chhota number, remainder bache" (LCM + r) Wo sabse chhota number jo 6, 8, 12 se divide karne par har baar remainder 5 de?
Step 1: LCM(6, 8, 12) = 24
Step 2: Remainder PLUS karo: 24 + 5 = 29 ✓ (29÷6 → rem 5 ✓)

✏️ Type 4 — "Sabse bada 4-digit number jo 15, 20, 25 se divide ho" Step 1: LCM(15, 20, 25) = 300
Step 2: Sabse bada 4-digit = 9999. Ise 300 se divide karo: 9999 ÷ 300 = 33, remainder 99
Step 3: 9999 − 99 = 9900 ✓ (ya 33 × 300 = 9900)

Trick 1: BADA-CHHOTA rule
BADA maango → HCF · CHHOTA maango → LCM
"Sabse bada number jo divide kare" = HCF. "Sabse chhota number jo divide ho" = LCM. Bas question ka keyword pakdo!

Trick 2: Remainder ka PLUS-MINUS rule
HCF wale mein MINUS · LCM wale mein PLUS
Sabse BADA + remainder bache → numbers me se remainder ghatao, phir HCF. Sabse CHHOTA + remainder bache → LCM nikaal ke remainder jodo.

Trick 3: Fractions ka ULTA-SEEDHA
HCF: upar-HCF / niche-LCM
HCF of fractions mein upar (numerator) ka HCF, niche (denominator) ka LCM. LCM of fractions mein bilkul ULTA. "Jo maanga hai wahi upar lagao!"

Trick 4: Co-prime shortcut
Co-prime: HCF = 1, LCM = product
13 aur 17 co-prime hain → HCF seedha 1, LCM seedha 13×17 = 221. Calculation ki zaroorat hi nahi!

Trick 5: Answer ka SANITY CHECK
HCF ≤ chhota number · LCM ≥ bada number
HCF kabhi sabse chhote number se bada nahi ho sakta, LCM kabhi sabse bade se chhota nahi. Option elimination mein ye 5-second check bahut MCQs bacha deta hai!

❌ Galti 1: HCF × LCM formula 3 numbers par lagana HCF × LCM = product sirf DO numbers ke liye sach hai. Teen numbers diye hon to ye formula seedha mat lagao!

❌ Galti 2: Fractions ka formula ulta laga dena HCF of fractions = HCF/LCM (upar HCF). Jaldi mein log LCM/HCF laga dete hain. Yaad rakho: "jo pucha hai, wahi UPAR".

❌ Galti 3: Remainder PLUS karna jahan MINUS chahiye "Sabse BADA number divide kare, remainder 2 bache" → numbers se 2 GHATAO phir HCF. "Sabse CHHOTA number, remainder 5" → LCM + 5. Dono types mix mat karo.

❌ Galti 4: Impossible HCF-LCM pair na pakad pana LCM hamesha HCF ka multiple hota hai. Agar option mein HCF = 8, LCM = 60 diya ho — ye possible hi nahi (60 ÷ 8 poora nahi)! Aisa question turant pakdo.

❌ Galti 5: Division method mein non-common prime bhool jana LCM nikalte waqt aakhri row ke SAARE numbers bhi multiply karne hote hain, sirf divisors nahi. 12, 18 → divisors 2, 3 aur bache hue 2, 3 → LCM = 2×3×2×3 = 36 (sirf 2×3 = 6 NAHI).

HCF ka full form?tap 👆

Highest Common Factor (Mahattam Samapvartak)

LCM ka full form?tap 👆

Lowest Common Multiple (Laghuttam Samapvartya)

HCF × LCM = ? (2 numbers)tap 👆

Dono numbers ka product

Co-prime numbers ka HCF?tap 👆

1 (aur LCM = product)

HCF of fractions = ?tap 👆

HCF(numerators) / LCM(denominators)

LCM of fractions = ?tap 👆

LCM(numerators) / HCF(denominators)

"Bells saath kab bajengi?" — kya lagau?tap 👆

LCM of intervals

"Sabse bada number jo divide kare" = ?tap 👆

HCF

"Sabse bada, remainder r bache" = ?tap 👆

HCF of (numbers − r)

"Sabse chhota, remainder r bache" = ?tap 👆

LCM + r

HCF prime-powers mein kaunsi power?tap 👆

Common primes ki sabse CHHOTI power

LCM prime-powers mein kaunsi power?tap 👆

Sab primes ki sabse BADI power

HCF kabhi kis se bada nahi hota?tap 👆

Sabse chhote number se (HCF ≤ smallest)

LCM kabhi kis se chhota nahi hota?tap 👆

Sabse bade number se (LCM ≥ largest)

LCM hamesha kiska multiple hota hai?tap 👆

HCF ka — warna pair impossible!

HCF(13, 17) aur LCM(13, 17)?tap 👆

HCF = 1, LCM = 13×17 = 221

Situation / Question	Formula / Action
2 numbers ka HCF × LCM	= dono ka product
Co-prime pair	HCF = 1, LCM = product
HCF of fractions	HCF(upar) / LCM(niche)
LCM of fractions	LCM(upar) / HCF(niche)
Bells / lights saath	LCM of intervals
Sabse bada divisor, remainder r	HCF of (numbers − r)
Sabse chhota number, remainder r	LCM + r
Sabse bada n-digit number divisible	Bade number mein se remainder ghatao
HCF ki limit	≤ sabse chhota number
LCM ki limit	≥ sabse bada number, HCF ka multiple

🎯 Score: 0 / 0 Total questions: 14

Practice (exam-style)

Q1. The HCF of 24 and 36 is:

💡 24 = 2³×3, 36 = 2²×3² → common ki chhoti power: 2²×3 = 12.

Practice (exam-style)

Q2. The LCM of 12 and 18 is:

💡 12 = 2²×3, 18 = 2×3² → badi powers: 2²×3² = 36.

Practice (exam-style)

Q3. The HCF and LCM of two numbers are 12 and 72. If one number is 24, the other number is:

💡 HCF×LCM = product → doosra = (12×72)÷24 = 36.

Practice (exam-style)

Q4. The HCF of 2/3 and 4/9 is:

💡 HCF(2,4)/LCM(3,9) = 2/9. Upar HCF, niche LCM!

Practice (exam-style)

Q5. Three bells ring at intervals of 6, 8 and 12 seconds. If they ring together now, after how many seconds will they ring together again?

💡 "Saath bajengi" = LCM(6,8,12) = 24 seconds.

Practice (exam-style)

Q6. The greatest number which divides 62, 132 and 237 leaving remainder 2 in each case is:

💡 Remainder minus karo: HCF(60, 130, 235) = 5.

Practice (exam-style)

Q7. The least number exactly divisible by 12, 15 and 20 is:

💡 "Sabse chhota divisible" = LCM(12,15,20) = 60.

Practice (exam-style)

Q8. The least number which when divided by 6, 8 and 12 leaves remainder 5 in each case is:

💡 LCM(6,8,12) = 24, phir remainder jodo: 24+5 = 29.

Practice (exam-style)

Q9. The LCM of two co-prime numbers 13 and 17 is:

💡 Co-prime ka LCM = product = 13×17 = 221 (HCF hota 1).

Practice (exam-style)

Q10. The LCM of 2/5 and 3/10 is:

💡 LCM(2,3)/HCF(5,10) = 6/5. Upar LCM, niche HCF!

Practice (exam-style)

Q11. Two numbers are in the ratio 3 : 4 and their HCF is 5. Their LCM is:

💡 Numbers = 15 aur 20 → LCM = 60 (ya shortcut: 3×4×HCF = 12×5).

Practice (exam-style)

Q12. Which pair of numbers has HCF 8 and LCM 48?

💡 16 = 2⁴, 24 = 2³×3 → HCF 8, LCM 48 ✓ (check: 8×48 = 16×24 = 384).

Practice (exam-style)

Q13. The greatest 4-digit number exactly divisible by 15, 20 and 25 is:

💡 LCM = 300; 9999 ÷ 300 → remainder 99; 9999−99 = 9900.

Practice (exam-style)

Q14. The product of two numbers is 216 and their HCF is 6. Their LCM is:

💡 LCM = product ÷ HCF = 216 ÷ 6 = 36 (jaise pair 6 aur 36).

HCF = sabse bada common divide karne wala, LCM = sabse chhota common table wala. Do numbers ke liye HCF × LCM = product (teen par nahi!). Word problems ka mantra: BADA maango → HCF, CHHOTA maango → LCM; remainder same bache to HCF wale mein minus, LCM wale mein plus. Fractions: HCF = upar-HCF/niche-LCM (LCM mein ulta). Sanity check hamesha: HCF ≤ smallest, LCM ≥ largest, aur LCM hamesha HCF ka multiple. Ye patterns ratt gaye to HCF-LCM ke saare MCQs 30 second wale ho jayenge! 🚀