Quick Reference Guide

Number System

Aptitude
Beginner Level

Number System

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May 23, 2026

About This Cheat Sheet

Hey there! Welcome to your complete guide to Number System for competitive exams. I've put together everything you need to know β€” from types of numbers (natural, whole, integers, rational, irrational, real, prime, composite, even, odd, co-prime) to divisibility rules (2 to 19), factors and multiples, LCM and HCF, remainders and modulus, unit digit concepts, number of trailing zeros, base system (binary, octal, hexadecimal), and important formulas with solved examples. Whether you're preparing for SSC, Banking, Railway, UPSC, or other government exams, this cheat sheet covers concepts and shortcut techniques to help you score better.

πŸ”’ Types of Numbers

  • Natural Numbers: 1, 2, 3, 4, ...
  • Whole Numbers: 0, 1, 2, 3, ...
  • Integers: ..., -3, -2, -1, 0, 1, 2, 3, ...
  • Rational Numbers: p/q form (q β‰  0)
  • Irrational Numbers: Not in p/q form
  • Real Numbers: Rational + Irrational
  • Prime Numbers: >1, only 2 factors
  • Composite Numbers: >1, >2 factors

πŸ“ Divisibility Rules

  • 2: Last digit even
  • 3: Sum of digits divisible by 3
  • 4: Last 2 digits divisible by 4
  • 5: Last digit 0 or 5
  • 6: Divisible by 2 and 3
  • 7: Double last digit, subtract
  • 8: Last 3 digits divisible by 8
  • 9: Sum of digits divisible by 9

πŸ“Š LCM & HCF

  • LCM: Least Common Multiple
  • HCF: Highest Common Factor
  • LCM Γ— HCF: = Product of numbers
  • Prime Factorization: Find factors
  • Division Method: For HCF
  • LCM of Fractions: LCM of numerators / HCF of denominators
  • HCF of Fractions: HCF of numerators / LCM of denominators
  • Applications: Time, speed, distance

πŸ” Remainders & Modulus

  • Remainder Theorem: Dividend = Divisor Γ— Quotient + Remainder
  • Negative Remainder: r = (d - n) where n > d
  • Modulus Operation: a mod b
  • Cyclicity of Remainders: Pattern repeats
  • Euler's Theorem: a^Ο†(n) ≑ 1 (mod n)
  • Fermat's Theorem: a^(p-1) ≑ 1 (mod p)
  • Chinese Remainder Theorem: Solve simultaneous congruences
  • Wilson's Theorem: (p-1)! ≑ -1 (mod p)

πŸ”Ÿ Unit Digit & Cyclicity

  • Cyclicity of 2: 2, 4, 8, 6
  • Cyclicity of 3: 3, 9, 7, 1
  • Cyclicity of 4: 4, 6
  • Cyclicity of 7: 7, 9, 3, 1
  • Cyclicity of 8: 8, 4, 2, 6
  • Cyclicity of 9: 9, 1
  • Unit Digit of a^b: Find pattern
  • Trailing Zeros: Count 5s in factorial

πŸ”’ Base System

  • Binary: Base 2 (0, 1)
  • Octal: Base 8 (0-7)
  • Decimal: Base 10 (0-9)
  • Hexadecimal: Base 16 (0-9, A-F)
  • Conversion: Any base to decimal
  • Decimal to Binary: Divide by 2
  • Decimal to Octal: Divide by 8
  • Decimal to Hex: Divide by 16

πŸ”’ Prime

Only 2 factors

2, 3, 5, 7...

πŸ“ Divisibility

2, 3, 4, 5, 6, 7, 8, 9

Quick rules

πŸ“Š LCM Γ— HCF

= Product of numbers

Important formula

πŸ”Ÿ Unit Digit

Cyclicity pattern

2, 3, 7, 8

πŸ”’ Types of Numbers

πŸ“Œ Number Type
  • Natural Numbers
  • Whole Numbers
  • Integers
  • Rational Numbers
  • Irrational Numbers
  • Real Numbers
  • Prime Numbers
  • Composite Numbers
  • Even Numbers
  • Odd Numbers
  • Co-prime Numbers
πŸ“ Definition
  • Counting numbers: 1, 2, 3, 4, ...
  • Natural + 0: 0, 1, 2, 3, ...
  • ..., -3, -2, -1, 0, 1, 2, 3, ...
  • Numbers in p/q form (q β‰  0)
  • Not in p/q form (√2, Ο€)
  • Rational + Irrational numbers
  • >1 with exactly 2 factors
  • >1 with more than 2 factors
  • Divisible by 2: 0, 2, 4, 6, ...
  • Not divisible by 2: 1, 3, 5, 7, ...
  • HCF = 1 (e.g., 2 & 3)
πŸ’‘ Examples
  • 1, 2, 3, 4, 5, ...
  • 0, 1, 2, 3, 4, ...
  • -2, -1, 0, 1, 2
  • 2/3, 5/7, 0.75
  • √2, √3, Ο€, e
  • All rational + irrational
  • 2, 3, 5, 7, 11, 13
  • 4, 6, 8, 9, 10, 12
  • 0, 2, 4, 6, 8, 10
  • 1, 3, 5, 7, 9, 11
  • 2 & 3, 4 & 5, 8 & 9

πŸ“ Divisibility Rules

πŸ”’ Rule
  • 2
  • 3
  • 4
  • 5
  • 6
  • 7
  • 8
  • 9
  • 10
  • 11
  • 13
  • 17
  • 19
πŸ“ Condition
  • Last digit is even (0, 2, 4, 6, 8)
  • Sum of digits divisible by 3
  • Last 2 digits divisible by 4
  • Last digit is 0 or 5
  • Divisible by 2 and 3
  • Double last digit, subtract from rest, divisible by 7
  • Last 3 digits divisible by 8
  • Sum of digits divisible by 9
  • Last digit is 0
  • Difference of (sum odd places - sum even places) divisible by 11
  • Four times last digit + rest, divisible by 13
  • Five times last digit - rest, divisible by 17
  • Two times last digit + rest, divisible by 19
πŸ’‘ Example
  • 246 β†’ 6 is even βœ“
  • 123 β†’ 1+2+3=6 βœ“
  • 124 β†’ 24 divisible by 4 βœ“
  • 125 β†’ 5 βœ“
  • 246 β†’ even + sum=12 βœ“
  • 133 β†’ 13 - 6 = 7 βœ“
  • 1240 β†’ 240 divisible by 8 βœ“
  • 153 β†’ 1+5+3=9 βœ“
  • 250 β†’ 0 βœ“
  • 121 β†’ (1+1) - 2 = 0 βœ“
  • 78 β†’ 8Γ—4+7=39 βœ“
  • 85 β†’ 5Γ—5-8=17 βœ“
  • 57 β†’ 7Γ—2+5=19 βœ“
⚠️ Note
  • Check last digit
  • Sum all digits
  • Check last 2 digits
  • Check last digit
  • Combine 2 and 3
  • Repeat if needed
  • Check last 3 digits
  • Sum all digits
  • Check last digit
  • Difference method
  • Four times + rest
  • Five times - rest
  • Two times + rest

πŸ“Š LCM & HCF

πŸ“Œ Concept
  • LCM
  • HCF
  • LCM Γ— HCF
  • LCM of Fractions
  • HCF of Fractions
πŸ“ Formula / Method
  • Least Common Multiple
  • Highest Common Factor
  • = Product of two numbers
  • = LCM (numerators) / HCF (denominators)
  • = HCF (numerators) / LCM (denominators)
πŸ’‘ Example
  • LCM(4,6) = 12
  • HCF(4,6) = 2
  • 4 Γ— 6 = 24 = 12 Γ— 2 βœ“
  • LCM(2/3, 4/5) = LCM(2,4) / HCF(3,5) = 4/1 = 4
  • HCF(2/3, 4/5) = HCF(2,4) / LCM(3,5) = 2/15

πŸ”Ÿ Unit Digit & Cyclicity

πŸ”’ Number
  • 2
  • 3
  • 4
  • 5
  • 6
  • 7
  • 8
  • 9
  • 0
  • 1
πŸ”„ Cyclicity
  • 2, 4, 8, 6 (4-cycle)
  • 3, 9, 7, 1 (4-cycle)
  • 4, 6 (2-cycle)
  • 5 (1-cycle)
  • 6 (1-cycle)
  • 7, 9, 3, 1 (4-cycle)
  • 8, 4, 2, 6 (4-cycle)
  • 9, 1 (2-cycle)
  • 0 (1-cycle)
  • 1 (1-cycle)
πŸ“ Pattern
  • 2ΒΉ=2, 2Β²=4, 2Β³=8, 2⁴=16(6)
  • 3ΒΉ=3, 3Β²=9, 3Β³=27(7), 3⁴=81(1)
  • 4ΒΉ=4, 4Β²=16(6)
  • 5ΒΉ=5, 5Β²=25(5)
  • 6ΒΉ=6, 6Β²=36(6)
  • 7ΒΉ=7, 7Β²=49(9), 7Β³=343(3), 7⁴=2401(1)
  • 8ΒΉ=8, 8Β²=64(4), 8Β³=512(2), 8⁴=4096(6)
  • 9ΒΉ=9, 9Β²=81(1)
  • Always 0
  • Always 1
πŸ’‘ Example
  • Unit digit of 2¹⁰⁰: 100 mod 4 = 0 β†’ 6
  • Unit digit of 3⁡⁰: 50 mod 4 = 2 β†’ 9
  • Unit digit of 4²⁡: 25 mod 2 = 1 β†’ 4
  • Unit digit of 5⁷⁡: 5
  • Unit digit of 6⁹⁹: 6
  • Unit digit of 7⁢⁰: 60 mod 4 = 0 β†’ 1
  • Unit digit of 8²⁰: 20 mod 4 = 0 β†’ 6
  • Unit digit of 9³⁰: 30 mod 2 = 0 β†’ 1
  • Unit digit of 10⁡: 0
  • Unit digit of 1⁡⁰: 1

πŸ“ Important Formulas

πŸ“Œ Formula
  • Sum of first n natural numbers
  • Sum of squares of first n natural numbers
  • Sum of cubes of first n natural numbers
  • Sum of first n even numbers
  • Sum of first n odd numbers
  • Product of n natural numbers
πŸ“ Formula
  • n(n+1)/2
  • n(n+1)(2n+1)/6
  • [n(n+1)/2]Β²
  • n(n+1)
  • nΒ²
  • n! (n factorial)
πŸ’‘ Example (n=5)
  • 5Γ—6/2 = 15
  • 5Γ—6Γ—11/6 = 55
  • (5Γ—6/2)Β² = 225
  • 5Γ—6 = 30
  • 25
  • 5! = 120

πŸ”Ÿ Trailing Zeros

πŸ“Œ Concept
  • Trailing Zeros in n!
  • Trailing Zeros in product
  • Method
πŸ“ Formula / Method
  • Number of zeros = floor(n/5) + floor(n/25) + floor(n/125) + ...
  • Count powers of 5 in factorization
  • Divide n by 5 repeatedly and sum quotients

πŸ”’ Base System

πŸ“Œ Base
  • Binary
  • Octal
  • Decimal
  • Hexadecimal
πŸ”’ Digits
  • 0, 1
  • 0, 1, 2, 3, 4, 5, 6, 7
  • 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
  • 0-9, A(10), B(11), C(12), D(13), E(14), F(15)
πŸ“ Base Value
  • Base 2
  • Base 8
  • Base 10
  • Base 16
πŸ’‘ Example
  • 1011β‚‚ = 1Γ—8+0Γ—4+1Γ—2+1 = 11₁₀
  • 17β‚ˆ = 1Γ—8+7 = 15₁₀
  • 123₁₀ = 123
  • A3₁₆ = 10Γ—16+3 = 163₁₀

πŸ”„ Conversion Between Bases

πŸ“Œ Conversion
  • Any Base β†’ Decimal
  • Decimal β†’ Binary
  • Decimal β†’ Octal
  • Decimal β†’ Hex
  • Binary β†’ Octal
  • Binary β†’ Hex
  • Octal β†’ Binary
  • Hex β†’ Binary
πŸ“ Method
  • Multiply each digit by base^position
  • Divide by 2 repeatedly
  • Divide by 8 repeatedly
  • Divide by 16 repeatedly
  • Group bits in 3s (octal)
  • Group bits in 4s (hex)
  • Each digit β†’ 3 bits
  • Each digit β†’ 4 bits
πŸ’‘ Example
  • 101₁₀ = 1Γ—4+0Γ—2+1 = 5₁₀
  • 13₁₀ β†’ 1101β‚‚
  • 15₁₀ β†’ 17β‚ˆ
  • 255₁₀ β†’ FF₁₆
  • 101101β‚‚ β†’ 55β‚ˆ
  • 101101β‚‚ β†’ 2D₁₆
  • 7β‚ˆ β†’ 111β‚‚
  • F₁₆ β†’ 1111β‚‚

πŸ”’ Important Number Systems

πŸ“Œ System
  • Natural Numbers
  • Whole Numbers
  • Integers
  • Rational Numbers
  • Irrational Numbers
  • Real Numbers
πŸ“ Includes
  • Positive integers only
  • Natural + 0
  • Positive + Negative + 0
  • p/q form (q β‰  0)
  • Non-terminating, non-repeating
  • Rational + Irrational
πŸ”’ Examples
  • 1, 2, 3, 4, ...
  • 0, 1, 2, 3, ...
  • ..., -3, -2, -1, 0, 1, ...
  • 1/2, 3/4, 0.25, 0.333...
  • √2, Ο€, e, √3
  • All rational + irrational
πŸ’‘ Key Point
  • Smallest natural = 1
  • Smallest whole = 0
  • No smallest integer
  • Can be terminating or repeating
  • Cannot be expressed as p/q
  • All numbers on number line

🎯 Number System Quick Revision

The Golden Rule: Understand the fundamentals β€” types of numbers, divisibility rules, and LCM/HCF are the building blocks. Master unit digit cyclicity for fast calculation in exams.

Quick Memory Tricks:

  • πŸ“Œ Divisibility: "2 Even, 3 Sum, 4 Last Two, 5 End, 6 Both 2&3, 9 Sum, 11 Difference"
  • πŸ“Œ Unit Digit Cyclicity: "2-4-8-6, 3-9-7-1, 4-6, 5-5, 6-6, 7-9-3-1, 8-4-2-6, 9-1, 0-0, 1-1"
  • πŸ“Œ LCM Γ— HCF: = Product of two numbers
  • πŸ“Œ Trailing Zeros: Count 5s in factorial
  • πŸ“Œ Base Conversion: Multiply by base powers

Exam Strategy:

  • Step 1: Identify the type of number problem
  • Step 2: Apply divisibility rules for quick checks
  • Step 3: Use unit digit cyclicity for large powers
  • Step 4: Apply LCM/HCF formulas for ratio problems
  • Step 5: Use base conversion for binary/hex questions

πŸ’‘ Pro Tips for Number System

βœ… Learn Divisibility Rules

Memorize rules for 2, 3, 4, 5, 6, 7, 8, 9, 11. These are frequently tested in competitive exams.

βœ… Master Unit Digit Cyclicity

Unit digit problems appear in almost every exam. Learn the cyclicity patterns of 2, 3, 4, 7, 8, 9.

βœ… LCM Γ— HCF Formula

For two numbers, LCM Γ— HCF = Product of numbers. This is extremely useful for quickly solving problems.

βœ… Trailing Zeros in Factorial

Always count powers of 5. Formula: floor(n/5) + floor(n/25) + floor(n/125) + ...

βœ… Base System Conversions

Practice converting between binary, octal, decimal, and hexadecimal. These are common in computer aptitude sections.

βœ… Practice Remainder Problems

Euler's Theorem, Fermat's Theorem, and Chinese Remainder Theorem are high-level concepts that appear in advanced exams.

πŸ“‹ Quick Reference: Divisibility Rules Summary

πŸ”’ Number
  • 2
  • 3
  • 4
  • 5
  • 6
  • 7
  • 8
  • 9
  • 10
  • 11
  • 13
  • 17
  • 19
πŸ“ Rule
  • Last digit even
  • Sum of digits divisible by 3
  • Last 2 digits divisible by 4
  • Last digit 0 or 5
  • Divisible by 2 and 3
  • Double last digit, subtract from rest
  • Last 3 digits divisible by 8
  • Sum of digits divisible by 9
  • Last digit 0
  • Difference of (odd-even sum) divisible by 11
  • 4 Γ— last digit + rest
  • 5 Γ— last digit βˆ’ rest
  • 2 Γ— last digit + rest
πŸ’‘ Check
  • 246 βœ“
  • 123 βœ“
  • 124 βœ“
  • 125 βœ“
  • 246 βœ“
  • 133 βœ“
  • 1240 βœ“
  • 153 βœ“
  • 250 βœ“
  • 121 βœ“
  • 78 βœ“
  • 85 βœ“
  • 57 βœ“
⚠️ Action
  • Check last digit
  • Sum digits
  • Check last 2
  • Check last digit
  • Combine 2 and 3
  • Repeat if needed
  • Check last 3
  • Sum digits
  • Check last digit
  • Find difference
  • 4Γ— + rest
  • 5Γ— βˆ’ rest
  • 2Γ— + rest

πŸ”Ÿ Quick Reference: Unit Digit Summary

πŸ”’ Number
  • 2
  • 3
  • 4
  • 5
  • 6
  • 7
  • 8
  • 9
  • 0
  • 1
πŸ”„ Cyclicity
  • 2, 4, 8, 6
  • 3, 9, 7, 1
  • 4, 6
  • 5
  • 6
  • 7, 9, 3, 1
  • 8, 4, 2, 6
  • 9, 1
  • 0
  • 1
πŸ“ Length
  • 4
  • 4
  • 2
  • 1
  • 1
  • 4
  • 4
  • 2
  • 1
  • 1
πŸ’‘ Example
  • 2⁡⁰ β†’ 50%4=2 β†’ 4
  • 3⁢⁰ β†’ 60%4=0 β†’ 1
  • 4²⁡ β†’ 25%2=1 β†’ 4
  • 5⁷⁡ β†’ 5
  • 6⁹⁹ β†’ 6
  • 7⁷⁰ β†’ 70%4=2 β†’ 9
  • 8⁴⁰ β†’ 40%4=0 β†’ 6
  • 9³⁰ β†’ 30%2=0 β†’ 1
  • 0⁡⁰ β†’ 0
  • 1⁡⁰ β†’ 1

Topics Covered

number-system quantitative-aptitude divisibility-rules lcm-hcf remainders unit-digit base-system competitive-exams ssc banking railway

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