CAT Quantitative Aptitude - Number System Complete Guide

Complete Number System guide for CAT Quantitative Aptitude:

1. CLASSIFICATION OF NUMBERS

   • Natural numbers, Whole numbers, Integers
   • Rational and Irrational numbers
   • Real and Complex numbers

2. DIVISIBILITY RULES

   • Rules for 2, 3, 4, 5, 6, 8, 9, 10, 11
   • Applications in problem solving

3. FACTORS AND MULTIPLES

   • Prime and composite numbers
   • Prime factorization
   • HCF and LCM
   • Number of factors

4. REMAINDER THEOREM

   • Basic remainder concepts
   • Cyclicity and patterns
   • Chinese remainder theorem

5. BASE SYSTEM

   • Conversion between different bases
   • Operations in different bases

6. UNIT DIGITS AND LAST TWO DIGITS

   • Patterns in unit digits
   • Cyclicity of powers
   • Last two digits of powers

Key Concepts:

• For any number N = p₁^a₁ × p₂^a₂ × ... × pₖ^aₖ
• Number of factors = (a₁+1)(a₂+1)...(aₖ+1)
• Sum of factors = [(p₁^(a₁+1)-1)/(p₁-1)] × ... × [(pₖ^(aₖ+1)-1)/(pₖ-1)]

Important Formulas:

• LCM × HCF = Product of two numbers
• 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)

Quick Tricks:

• Divisibility by 11: Alternating sum of digits
• Finding last digit of aⁿ: Use cyclicity patterns
• HCF by Euclid's algorithm

Practice Sets:

• 200+ practice questions
• Previous CAT questions (2010-2023)
• Difficulty levels: Easy, Medium, Hard
• Time-bound practice tests

Shortcuts and Techniques:

• Mental math techniques
• Approximation methods
• Elimination strategies