noteswale

Number System

SSC CGL • Tier-I Quant • Topic 1 of 18 • ~45 min padhne + ~30 min practice

HAR SAAL 2-3 QUESTIONSWEIGHTAGE: HIGH

Hindi mode me: questions pure Hindi me • notes & tricks Hinglish me (teacher voice)

Poora topic ek line me

Numbers ke saath khelna seekhna hai — kaun kisse kat-ta hai (divisibility), common factors (HCF-LCM), aur bachaa hua hissa (remainder). SSC inhi teen cheezon ko ghuma-phira ke puchhta hai.

30 second ka waada: neeche wali table yaad kar lo, aadha topic khatm.
Kuch khaas nahi — ye Quant ka pehla topic hai, yahin se shuruaat hoti hai. Bas basic jod-ghata aana chahiye.
HCF-LCM se ~11 questions, divisibility se ~6, remainders se ~5. 90% questions formula ya trick-direct the — lamba calculation wala ek bhi nahi.
1. HCF × LCM = product (har saal aata hai)   2. Divisibility rules 2, 3, 11   3. Unit digit cyclicity. Bas — 80% marks yahin se.
  1. Divisibility Rules bina divide kiye check — har saal 1 question pakka
  2. HCF & LCM sabse zyada questions yahin se
  3. Remainders chhota topic, par 1 question fix
  4. Square Roots & Cubes basic, jaldi khatm
  5. Number Properties even/odd, prime, unit digit

Bina divide kiye ye check karna ki koi number divisible hai ya nahi. SSC me har saal 1 question pakka.

Se divideRule
2Last digit even ho (0, 2, 4, 6, 8)
3Digits ka sum 3 se divisible ho
4Last 2 digits 4 se divisible hon
5Last digit 0 ya 5 ho
8Last 3 digits 8 se divisible hon
9Digits ka sum 9 se divisible ho
11Alternate digits ke sum ka difference 0 ya 11 ka multiple ho
62 aur 3 dono se divisible ho
10Last digit 0 ho
123 aur 4 dono se divisible ho
YAAD RAKHO3 aur 9 ka rule same family ka hai — digits ke sum se. 11 ka rule alag hai — alternate digits se. Dono ko mix mat karna.
SHORTCUT6, 10, 12 ke rule alag se mat rato — co-prime factors me todo. 6 = 2×3, 12 = 3×4. Dono se kat-ta hai to usse bhi kat-ta hai.

Even/Odd: 0 even hota hai (na even-na odd wali baat galat hai — neeche mythbust dekho).

Prime numbers: 1 prime nahi hota. 2 sabse chhota (aur akela even) prime hai. 1 se 20 tak 8 primes: 2, 3, 5, 7, 11, 13, 17, 19.

Co-prime: jinka HCF 1 ho — jaise 8 aur 15. Dono prime hona zaroori nahi!

FORMULA1 + 2 + … + n = n(n+1)/2

Is topic me ye squares yaad hone chahiye. Shuru karne se pehle ek baar bol ke dekho:

11²=121, 12²=144, 13²=169, 14²=196, 15²=225, 16²=256, 17²=289, 18²=324, 19²=361, 20²=400

25²=625, 30²=900. Cubes: 2³=8, 3³=27, 4³=64, 5³=125, 10³=1000.

CHECKPOINTEASY
Check: Is 7,722 divisible by 11?
Yes
No
EXPLANATION (HINGLISH)Alternate digits: (7+2) − (7+2) = 9 − 9 = 0 → divisible. Yes. Agar ye samajh aaya to Layer 3 pe chalo.

HCF & LCM

HCF (Highest Common Factor • महत्तम समापवर्तक): sabse bada number jo sabko divide kare. LCM (Lowest Common Multiple • लघुत्तम समापवर्त्य): sabse chhota number jo sabse divide ho jaye.

Method: prime factorisation karo → HCF me common factors ki lowest power, LCM me sabhi factors ki highest power.

BEGINNER — FACTOR KARNA KAISE HAI48 ko todo jab tak sirf prime na bache: 48 = 2 × 24, 24 = 2 × 12, 12 = 2 × 6, 6 = 2 × 3. Ruko! Ab 48 = 2⁴ × 3. Neeche tree dekho — yehi prime factorisation hai.
48 2 24 2 12 2 6 2 3
48 = 2 × 2 × 2 × 2 × 3 = 2⁴ × 3 — bhare hue circle prime hain, unhe aage todo mat

Ladder method (exam me tez): 12, 18, 24 ka LCM — common number se divide karte jao: 2 se → 6, 9, 12; phir 2 se → 3, 9, 6; phir 3 se → 1, 3, 2. Ab koi common divisor nahi → LCM = 2 × 2 × 3 × 1 × 3 × 2 = 72. Factorisation se tez hai!

Example: 24, 36, 48 ka HCF? 24=2³×3, 36=2²×3², 48=2⁴×3 → lowest powers: 2²×3 = 12.

COMMON MISTAKEHCF me lowest power, LCM me highest power — ulta kar diya to answer galat. Yehi sabse common error hai.
11 ka 10-Second Rule SAFE 10 SEC
KAB KAAM KAREGA Jab check karna ho ki number 11 se divisible hai ya nahi.
TRICK Alternate digits lo: (1st+3rd+5th…) − (2nd+4th…). Difference 0 ya 11 ka multiple → divisible.
EXAMPLE 918082 → (9+8+8)−(1+0+2) = 25−3 = 22 → 11 ka multiple → divisible.
KAB FAIL HOGA: bade numbers me alternate digits ginne me galti hoti hai — ek digit chhoot gaya to answer ulta. Slow down, digits ko ungli se gino.
PYQ PROOF 2021 Shift 2, 2023 Shift 3 — dono me direct aaya tha.
HCF × LCM = Product SAFE 15 SEC
KAB KAAM KAREGA Jab 2 numbers me se HCF, LCM ya product me se koi 2 diye hon, teesra nikalna ho.
TRICK LCM = Product ÷ HCF. Bas.
EXAMPLE HCF=12, product=1800 → LCM = 1800÷12 = 150.
KAB FAIL HOGA: ye formula sirf 2 numbers ke liye hai! 3 numbers pe lagaya to galat answer. 3 numbers me HCF×LCM = product nahi hota.
PYQ PROOF 2019, 2022 — har 2-3 saal me ek baar pakka.
Unit Digit Wala Jugaad SAFE 15 SEC
KAB KAAM KAREGA Jab badi power ka unit digit puchha jaye (jaise 7²⁴³ ka unit digit).
TRICK Cyclicity yaad rakho — 2,3,7,8: 4 ka cycle; 4,9: 2 ka cycle; 0,1,5,6: hamesha same. Power ko cycle se divide karo, remainder dekho.
EXAMPLE 7²⁴³: 7 ka cycle 4 → 243÷4 remainder 3 → 7³=343 → unit digit 3.
KAB FAIL HOGA: remainder 0 aaye to cycle ka last lo, pehla nahi. 7⁴ ka unit digit 1 hai (7¹=7 nahi). Yehi trap hai.
PYQ PROOF 2020 Shift 1, 2024 Shift 2.
LCM − k Wala Pattern SAFE 20 SEC
KAB KAAM KAREGA Jab alag-alag numbers se divide karne pe alag-alag remainder bache, par (divisor − remainder) sab me same ho (= k).
TRICK Answer = LCM(divisors) − k.
EXAMPLE 5, 6, 7 se remainder 2, 3, 4 → 5−2=3, 6−3=3, 7−4=3 → k=3 → LCM(5,6,7) − 3 = 210 − 3 = 207.
KAB FAIL HOGA: agar (divisor − remainder) same na ho to ye trick mat lagao — wahan simple "LCM + remainder" wala case ho sakta hai. Pehle difference check karo!
PYQ PROOF PYQ pattern — har 2-3 saal me is family ka question aata hai.
HCF of Differences SAFE 20 SEC
KAB KAAM KAREGA "Sabse bada number jo a, b ko divide karke remainder r chhode" type questions.
TRICK Pehle remainder ghatao, phir HCF nikalo: HCF(a−r, b−r).
EXAMPLE 122, 243 me se remainder 2, 3 → 120, 240 → HCF = 120.
KAB FAIL HOGA: seedha 122, 243 ka HCF nikaloge to galat! Pehle remainder subtract karna compulsory hai — yehi examiner ka trap hai.
PYQ PROOF PYQ pattern — "greatest number" wale questions me pakka.
Co-prime Pairs Ginna SAFE 30 SEC
KAB KAAM KAREGA "Sum S hai, HCF h hai — kitne pairs possible?" type questions.
TRICK Numbers = h×x, h×y → x + y = S ÷ h. Ab x, y co-prime hone chahiye — aise pairs gino.
EXAMPLE Sum 36, HCF 4 → x+y = 9 → (1,8), (2,7), (4,5) = 3 pairs. (3,6) co-prime nahi — mat gino!
KAB FAIL HOGA: co-prime condition bhool gaye to extra pairs gin loge. Har pair ka HCF 1 hona chahiye — check karo.
PYQ PROOF 2021 Shift 2 (PYQ 0022) — direct aaya tha.
Remainder Cyclicity Funda RISKY 20 SEC
KAB KAAM KAREGA Jab aⁿ ÷ b me remainder nikalna ho aur a, b chhote hon.
TRICK a ko b ke mod me chhota karo (jaise 7 mod 4 = 3, ya −1). Phir power ka pattern dekho.
EXAMPLE 7³ ÷ 4: 7 ≡ −1 (mod 4) → (−1)³ = −1 → +4 = 3.
KAB FAIL HOGA: jab base aur divisor co-prime na hon (jaise 6ⁿ ÷ 4) — tab ye shortcut toot-ta hai, direct division karo. Pehle co-prime check karo!
PYQ PROOF 2022 Shift 3 (pattern).
7 9 3 1
7 ka cycle: 7 → 9 → 3 → 1 → 7 (4 ka cycle). Power ÷ 4 karo, remainder se digit pakdo.
VIRAL MYTH — PARDAFASH

Myth: "0 na even hai na odd hota hai."
Sach: 0 even hai — kyunki 0 = 2 × 0, aur jo number 2 se poora kat jaye wo even. SSC ne is confusion pe question puchha hai. Ab kabhi mat bhoolna.

Remainders

Basic rule: remainder hamesha divisor se chhota hota hai. Negative remainder ko positive me badlo: −1 mod 7 = 6.

≡ ka matlab ("mod me barabar"): 2 ≡ −1 (mod 3) ka matlab — 2 aur −1, dono ko 3 se divide karne pe same remainder milta hai. Isliye power wale questions me 2 ki jagah −1 likh sakte ho, calculation aasaan ho jati hai.

Example: 2¹⁰⁰ ÷ 3 me remainder? 2 ≡ −1 (mod 3) → (−1)¹⁰⁰ = 1 → remainder 1.

EXAMINER KI CHAAL

Examiner ko pata hai tum "HCF × LCM = product" rat loge. Isliye wo 3 numbers dega jahan ye formula kaam nahi karta — aur option me wahi galat answer rakhega jo formula se aayega. Trick yaad hai? Condition bhi yaad rakho.

Bina Trick vs Trick — farak dekho

BINA TRICK (60 sec)
7²⁴³ ka unit digit: poora multiply? Impossible. Log calculator uthate hain — exam me fail.
TRICK SE (15 sec)
Cycle 4 → 243÷4, remainder 3 → 7³=343 → 3. Ho gaya.

Square Roots & Cubes

Unit digit trick: square ka unit digit kabhi 2, 3, 7, 8 nahi hota. Option me ye dikhe to turant eliminate.

Example: √2025? 45²=2025 (4×5=20, 5²=25 — 45 ka shortcut). Answer 45.

SHORTCUT — 5 SE END HONE WALEn5² = n×(n+1) ke baad 25 lagao. 35² = 3×4=12 → 1225.

HCF vs LCM — confuse mat hona

HCFLCM
MatlabSabse bada common divisorSabse chhota common multiple
PowerLowestHighest
Hamesha≤ sabse chhota number≥ sabse bada number
Kab"divide karna", "baantna""repeat hoga", "saath me"
Question padho kya nikalna hai? baantna? repeat? tukde / baantna HCF saath / repeat LCM
Confusion ho to ye puchho: baantna hai → HCF, repeat / saath me → LCM
CHECKPOINTMEDIUM
Check: HCF × LCM = product works for which of these?
Three numbers: 4, 6, 8
Two numbers only
EXPLANATION (HINGLISH)Formula sirf 2 numbers ke liye hai — yehi examiner ki chaal wala point tha. Samajh aaya to practice pe chalo.

Advanced Formulas — PYQ me aate hain

YAAD RAKHO1² + 2² + … + n² = n(n+1)(2n+1)/6
YAAD RAKHO1³ + 2³ + … + n³ = [n(n+1)/2]²

Divisors: N = p^a × q^b ho to divisors ki sankhya = (a+1)(b+1). Jaise 72 = 2³×3² → (3+1)(2+1) = 12 divisors.

Recurring decimals: 0.333… = 1/3, 0.1666… = 1/6, 0.999… = 1. (0.999… = 1 hota hai — yehi examiner ka favourite trap hai!)

Top 3 Galtiyan Is Topic Me

1. HCF × LCM = product ko 3 numbers pe lagana. Sirf 2 numbers ke liye hai — examiner yahi phasata hai.

2. Unit digit me remainder 0 ka matlab. 7⁴ ka unit digit 1 hai, 7 nahi — remainder 0 = cycle ka last digit.

3. Divisible vs remainder. 5 se divide karne pe remainder 3 ka matlab number = 5k+3 — seedha LCM+3 lagao.

KAB CHHODNA HAI (−0.50 YAAD RAKHO)Easy: 45 sec se zyada lage to chhodo • Medium: 60 sec • Hard: 90 sec.

Shuru karne se pehle — 10 second

Topic pakka kab maanoge? Neeche 25 me se 20 sahi = pakka.

PRACTICEEASYCGL-Q-NS-0001
Which of the following numbers is divisible by 3?
A) 12344
B) 12345
C) 12346
D) 12347
PRACTICEEASYCGL-Q-NS-0002
The HCF of 12 and 18 is:
A) 3
B) 6
C) 9
D) 36
PRACTICEEASYCGL-Q-NS-0003
The LCM of 4 and 6 is:
A) 2
B) 10
C) 12
D) 24
PRACTICEEASYCGL-Q-NS-0004
The unit digit of 7² is:
A) 7
B) 9
C) 3
D) 1
PRACTICEEASYCGL-Q-NS-0005
The sum of the first 10 natural numbers is:
A) 45
B) 55
C) 50
D) 65
PRACTICEEASYCGL-Q-NS-0006
Is 0 an even number or an odd number?
A) Even
B) Odd
C) Neither
D) Both
PRACTICEMEDIUMCGL-Q-NS-0007
The HCF of 24, 36 and 48 is:
A) 6
B) 12
C) 24
D) 144
PRACTICEMEDIUMCGL-Q-NS-0008
The LCM of 12, 18 and 24 is:
A) 36
B) 48
C) 72
D) 144
PRACTICEMEDIUMCGL-Q-NS-0009
If the HCF of two numbers is 12 and their product is 1800, then their LCM is:
A) 140
B) 150
C) 160
D) 120
PRACTICEMEDIUMCGL-Q-NS-0010
What is the remainder when 7³ is divided by 4?
A) 1
B) 2
C) 3
D) 0
PRACTICEMEDIUMCGL-Q-NS-0011
Which of the following numbers is divisible by 11?
A) 1331
B) 1332
C) 1352
D) 1341
PRACTICEMEDIUMCGL-Q-NS-0012
The smallest number divisible by 12, 18 and 24 is:
A) 36
B) 72
C) 144
D) 216
PRACTICEMEDIUMCGL-Q-NS-0013
The value of √625 is:
A) 15
B) 35
C) 25
D) 125
PRACTICEMEDIUMCGL-Q-NS-0014
How many prime numbers are there between 1 and 20?
A) 7
B) 8
C) 9
D) 6
PRACTICEMEDIUMCGL-Q-NS-0015
The smallest number which when divided by 5 and 6 leaves remainder 3 in each case is:
A) 30
B) 33
C) 63
D) 27
PRACTICEMEDIUMCGL-Q-NS-0016
The unit digit of 3¹⁰ is:
A) 1
B) 3
C) 7
D) 9
PRACTICEHARDCGL-Q-NS-0017
The HCF of 2³ × 3² and 2² × 3³ × 5 is:
A) 72
B) 36
C) 18
D) 180
PRACTICEHARDCGL-Q-NS-0018
What is the remainder when 2¹⁰⁰ is divided by 3?
A) 1
B) 2
C) 0
D) Cannot be determined
PRACTICEHARDCGL-Q-NS-0019
The largest 4-digit number divisible by 12 is:
A) 9999
B) 9984
C) 9996
D) 9900
PRACTICEHARDCGL-Q-NS-0020
What is the remainder when 5⁵ is divided by 6?
A) 1
B) 2
C) 4
D) 5
PRACTICEMEDIUMCGL-Q-NS-0029
The sum 1² + 2² + ... + 10² is:
A) 375
B) 385
C) 395
D) 365
PRACTICEMEDIUMCGL-Q-NS-0030
The number of divisors of 72 is:
A) 8
B) 10
C) 12
D) 24
PRACTICEMEDIUMCGL-Q-NS-0031
The least number which when divided by 5, 6 and 7 leaves remainders 2, 3 and 4 respectively is:
A) 207
B) 210
C) 213
D) 204
PRACTICEMEDIUMCGL-Q-NS-0032
The greatest number which divides 122 and 243 leaving remainders 2 and 3 respectively is:
A) 60
B) 120
C) 240
D) 30
PRACTICEEASYCGL-Q-NS-0033
Which of the following numbers is divisible by 12?
A) 1234
B) 1235
C) 1236
D) 1237
PYQ 2022 • SHIFT 1MEDIUMCGL-Q-NS-0021
The LCM of 2/3, 4/9 and 5/6 is:
A) 20/3
B) 10/3
C) 5/3
D) 40/3
PYQ 2021 • SHIFT 2HARDCGL-Q-NS-0022
The sum of two numbers is 36 and their HCF is 4. How many such pairs are possible?
A) 2
B) 3
C) 4
D) 5
PYQ 2020 • SHIFT 3MEDIUMCGL-Q-NS-0023
The unit digit of 1! + 2! + 3! + ... + 50! is:
A) 1
B) 3
C) 0
D) 9
PYQ 2023 • SHIFT 1MEDIUMCGL-Q-NS-0024
If the number 7x3542 is divisible by 11, then the value of x is:
A) 5
B) 6
C) 7
D) 8
PYQ 2019 • SHIFT 2MEDIUMCGL-Q-NS-0025
The HCF of two numbers is 15 and their LCM is 180. If one number is 45, the other number is:
A) 50
B) 60
C) 75
D) 90
PYQ 2024 • SHIFT 1MEDIUMCGL-Q-NS-0026
How many numbers between 100 and 300 are divisible by 7?
A) 26
B) 28
C) 30
D) 29
PYQ 2018 • SHIFT 2MEDIUMCGL-Q-NS-0027
The smallest number to be subtracted from 1000 to make it a perfect square is:
A) 29
B) 39
C) 49
D) 19
PYQ PATTERNHARDCGL-Q-NS-0028
What is the remainder when 4¹⁰⁰⁰ is divided by 7?
A) 1
B) 2
C) 4
D) 6

Formula Sheet

Shabdavali

HCF (Highest Common Factor) — महत्तम समापवर्तक
Sabse bada number jo sabko divide kare.
LCM (Lowest Common Multiple) — लघुत्तम समापवर्त्य
Sabse chhota number jo sabse divide ho.
Divisible — विभाज्य
Poora kat jaye, remainder 0.
Remainder — शेषफल
Divide karne pe bachaa hissa.
Co-prime — सह-अभाज्य
Jinka HCF 1 ho (dono prime hona zaroori nahi).
Perfect square — पूर्ण वर्ग
Kisi number ka square, jaise 961 = 31².

Ye Topic Aage Kahan Kaam Aayega

Simplification → fraction simplify karne me HCF kaam aayega Ratio & Proportion → ratio ko simplest form me HCF se karte hain Percentage → fraction-percentage conversion isi pe based hai
NEXT TOPIC →
Simplification
BODMAS • Indices • Surds • 2-3 Qs har saal
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Number System Mini-Mock

15 Questions12 min−0.50 marking
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