📘 Number System Complete Cheat Sheet for JKSSB & SSC

A one-stop guide to master Number System for all competitive exams — JKSSB, SSC, Railway, Banking, and State PSCs.

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🧩 1. Basics of Number System

Type	Definition	Examples
Natural Numbers (N)	Counting numbers starting from 1	1, 2, 3, …
Whole Numbers (W)	Natural numbers + Zero	0, 1, 2, 3, …
Integers (Z)	Positive + Negative + Zero	-3, -2, -1, 0, 1, 2, 3
Rational Numbers (Q)	Can be expressed as p/q	½, -¾
Irrational Numbers	Cannot be expressed as p/q	√2, π
Real Numbers (R)	All Rational + Irrational	√2, 5, 1.2
Prime Numbers	Divisible only by 1 and itself	2, 3, 5, 7, 11
Composite Numbers	More than two factors	4, 6, 8, 9

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🔢 2. Divisibility Rules (Super Shortcut Table)

Number	Divisibility Rule
2	Last digit even (0,2,4,6,8)
3	Sum of digits divisible by 3
4	Last two digits divisible by 4
5	Last digit 0 or 5
6	Divisible by both 2 and 3
7	Double the last digit, subtract from remaining number → result divisible by 7
8	Last three digits divisible by 8
9	Sum of digits divisible by 9
10	Last digit 0
11	Difference between sum of odd and even position digits divisible by 11

🧠 Trick:
👉 If a number is divisible by 18, it must be divisible by 2 and 9.
👉 For 12, check divisibility by 3 and 4.

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💥 3. HCF & LCM Shortcuts

🔹 HCF (Greatest Common Divisor)

• Prime Factorization Method: Find common primes, multiply lowest powers.
• Shortcut:
  HCF × LCM = Product of two numbers

🔹 LCM

• Multiply all primes with highest power.

Example:
HCF & LCM of 12 and 18
👉 12 = 2² × 3¹
👉 18 = 2¹ × 3²
➡ HCF = 2¹ × 3¹ = 6
➡ LCM = 2² × 3² = 36
✅ Check: 6 × 36 = 12 × 18 = 216

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⚡ 4. Remainder & Modulo Tricks

🔹 Basic Rule:

a ≡ b (mod m) means (a - b) is divisible by m.

🔹 Shortcut Tricks:

Case	Rule	Example
Large powers	Find cyclicity of last digit	7¹ =7, 7²=49(9), 7³=343(3), 7⁴=2401(1) → cycle 7,9,3,1
Division remainder	(a+b) mod m = (a mod m + b mod m) mod m	(26+39) mod 5 = (1+4) mod 5 = 0
Negative remainder	Add divisor to make it positive	(-3) mod 7 = 4

🧠 Shortcut Example:
Find remainder when 7¹⁰⁰ is divided by 10
→ Cyclicity of 7 = (7,9,3,1) → length = 4
→ 100 mod 4 = 0 → 4th term = 1 ✅

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🧮 5. Unit Digit & Last Digit Tricks

🔹 Shortcut Steps:

1. Find the cyclicity of last digit.
2. Find power mod cycle length.

Base	Cycle	Length
2	2,4,8,6	4
3	3,9,7,1	4
4	4,6	2
7	7,9,3,1	4
8	8,4,2,6	4
9	9,1	2

✅ Example:
Find unit digit of 9³⁵ → cycle (9,1) → odd power → 9

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🧱 6. Factorization Shortcuts

🔹 Number of Factors

If N = a^p × b^q × c^r, then
👉 Total factors = (p+1)(q+1)(r+1)

🔹 Sum of Factors

👉 Sum = [(a^(p+1)−1)/(a−1)] × [(b^(q+1)−1)/(b−1)] × …

Example:

N = 36 = 2² × 3²
Factors = (2+1)(2+1)=9
Sum = [(2³−1)/(2−1)]×[(3³−1)/(3−1)] = 7×13=91 ✅

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🧠 7. Perfect Squares & Cubes Tricks

Observation	Shortcut
Last digit of square can’t be 2,3,7,8	So if number ends with these → not perfect square
Square of number ending in 5 → ends with 25	e.g., 25²=625
Cube of even = even; Cube of odd = odd	Simple elimination trick

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📏 8. Place Value & Face Value

Term	Meaning	Example
Face Value	Actual value of digit	In 345, FV of 4 = 4
Place Value	Digit × place	In 345, PV of 4 = 40

🧠 Tip:

• For quick subtraction type questions, write expanded form once, then eliminate same places.

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🧮 9. Conversion Tricks

Conversion	Formula
Binary → Decimal	Multiply each digit by 2ⁿ (right to left)
Decimal → Binary	Divide by 2, write remainders reverse
Octal ↔ Decimal	Multiply digits by 8ⁿ
Hex ↔ Decimal	Multiply digits by 16ⁿ

✅ Example:
Binary 1011 = (1×8)+(0×4)+(1×2)+(1×1)=11

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🧠 10. Miscellaneous Quick Tricks

• Sum of first n natural numbers: n(n+1)/2
• Sum of first n odd numbers: n²
• Sum of first n even numbers: n(n+1)
• Sum of squares (1²+2²+...+n²): n(n+1)(2n+1)/6
• Sum of cubes: [n(n+1)/2]²

✅ Example:
Sum of first 10 even numbers = 10×11=110

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🧮 11. Important Conceptual Differences

Concept	Shortcut Understanding
Prime	Only 2 divisors
Co-prime	HCF = 1 (not necessarily prime)
Perfect Number	Sum of divisors = number (6, 28)
Composite	More than 2 factors
Twin Primes	Difference = 2 (e.g. 11,13)

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🧠 12. Smart Solving Strategies (Exam-Focused)

1. Always eliminate options using unit digit logic first.
2. Avoid full multiplication — use divisibility + remainder tricks.
3. Convert ratios or fractions to decimals mentally when comparing.
4. Don’t calculate LCM fully if ratio-based. Work with factor logic.
5. Time-Saver: Learn cyclicity patterns (2,3,4,7,8,9) by heart.

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⚔️ 13. Practice Challenge Set (Quick Warm-Up)

1. Find remainder when 17⁵ is divided by 10.
2. Unit digit of (13)⁷⁸.
3. Find HCF and LCM of 72 and 108.
4. How many factors does 120 have?
5. Find sum of first 20 even numbers.
6. Check divisibility of 378 for 11.
7. Which number is co-prime with 45?