Reasoning Chapters

1. NUMBER AND LETTER SERIES2. ALPHANUMERIC SERIES3. MISSING NUMBER4. ANALOGY SIMILARTY5. CLASSIFICATION6. DIRECTION 7. BLOOD RELATION8. SEATING ARRANGEMENT9. PUZZLE10. ORDER AND RANKING11. SYLLOGISM12. CLOCK13. CALENDER14. WORD FORMATION15. DICE16. CUBE AND CUBOID17. MIRROR AND WATER IMAGE18. VENN DIAGRAM19. STATEMENT AND ASSUMPTION

Number System

“Number System is foundational in quantitative aptitude. Questions include types of numbers, divisibility, LCM-HCF, remainders, and more.”

1. Basic Concepts and Formulas

  • Natural Numbers: 1, 2, 3, …
  • Whole Numbers: 0, 1, 2, 3, …
  • Integers: …, -3, -2, -1, 0, 1, 2, …
  • Rational Numbers: Numbers of form p/q where q ≠ 0
  • Irrational Numbers: Cannot be expressed as p/q (e.g., √2, π)
  • Prime Numbers: Greater than 1 and divisible only by 1 and itself
  • Divisibility Rules: Check for 2, 3, 4, 5, 6, 9, 10 easily
  • LCM × HCF = Product of two numbers
  • Even Numbers: Divisible by 2 | Odd Numbers: Not divisible by 2
  • Sum of first n natural numbers = n(n + 1)/2

2. Type-Wise Questions

  • Type 1: Classification of Numbers
    Example: Is √49 rational? ⇒ Yes (value = 7)
  • Type 2: Divisibility Tests
    Example: Is 123456 divisible by 3? ⇒ Yes (sum of digits = 21)
  • Type 3: Prime Number Check
    Example: Is 29 a prime number? ⇒ Yes
  • Type 4: Find HCF and LCM
    Example: HCF(12, 18) = 6, LCM = 36
  • Type 5: LCM × HCF = Product Rule
    Example: If HCF = 4, LCM = 60 ⇒ Numbers = 12 & 20
  • Type 6: Remainder Theorem
    Example: What is remainder when 2⁵ is divided by 3? ⇒ 32 ÷ 3 = 2
  • Type 7: Even/Odd Operations
    Example: Even + Even = Even, Odd × Odd = Odd
  • Type 8: Sum of Series
    Example: Sum of 1st 20 natural numbers ⇒ 20×21/2 = 210
  • Type 9: Digital Root/9 Test
    Example: 846 → 8+4+6 = 18 ⇒ 1+8 = 9 ⇒ Divisible by 9
  • Type 10: Unit Digit Calculation
    Example: Unit digit of 7^4 = 2401 ⇒ Unit digit = 1

3. Previous Year Exam Questions

  • (SSC CGL 2022): What is the remainder when 123456 is divided by 9?
    Solution: Sum of digits = 21 ⇒ 21 ÷ 9 ⇒ Remainder = 3
  • (RRB NTPC 2020): If LCM of two numbers is 120 and their HCF is 4, what is the product of the numbers?
    Solution: Product = LCM × HCF = 120 × 4 = 480
  • (SSC CHSL 2019): Which of the following is a prime number? (a) 91 (b) 87 (c) 89 (d) 77
    Solution: (c) 89 is prime
  • (HSSC 2021): Find the smallest number which when divided by 5, 6, and 8 leaves remainder 1 in each case.
    Solution: LCM(5,6,8) = 120 ⇒ 120 + 1 = 121

    • 2. Advanced Types of Questions

    • Type 11: Number of Zeros in Factorial
      Example: Find the number of zeros at the end of 100!.
      Solution: Count the factors of 5 in 100!
      100 ÷ 5 = 20
      100 ÷ 25 = 4
      Total = 20 + 4 = 24 zeros
    • Type 12: Number of Divisors
      Example: Find the number of divisors of 36.
      Solution: Prime factorization of 36 = 2² × 3²
      Number of divisors = (2 + 1)(2 + 1) = 9 divisors
    • Type 13: Co-prime Numbers
      Example: Are 35 and 64 co-prime?
      Solution: HCF(35, 64) = 1, hence they are co-prime
    • Type 14: Remainder Theorem
      Example: Find the remainder when 123456 is divided by 9.
      Solution: Sum of digits = 1 + 2 + 3 + 4 + 5 + 6 = 21
      21 ÷ 9 gives remainder 3
    • Type 15: Digital Root
      Example: Find the digital root of 987654321.
      Solution: Sum of digits = 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45
      4 + 5 = 9, hence digital root is 9
    • Type 16: Number of Digits in a Power
      Example: Find the number of digits in 2¹⁰.
      Solution: Number of digits = ⌊log₁₀(2¹⁰)⌋ + 1
      log₁₀(2¹⁰) = 10 × log₁₀(2) ≈ 10 × 0.3010 = 3.010
      Number of digits = ⌊3.010⌋ + 1 = 4 digits
    • Type 17: Base Conversion
      Example: Convert 1011₂ to decimal.
      Solution: 1011₂ = 1×2³ + 0×2² + 1×2¹ + 1×2⁰ = 8 + 0 + 2 + 1 = 11₁₀
    • Type 18: Number of Perfect Squares
      Example: How many perfect squares are there between 1 and 100?
      Solution: Perfect squares are 1², 2², ..., 10². Total = 10
    • Type 19: Odd and Even Powers
      Example: Is 2⁵ odd or even?
      Solution: 2 raised to any positive integer power is even, hence 2⁵ is even
    • Type 20: LCM and HCF Relationship
      Example: If LCM(12, 15) = 60 and HCF(12, 15) = 3, verify the relationship LCM × HCF = Product of
    • Type 21: Highest Power of a Prime in Factorial
      Example: Highest power of 2 in 50!
      50/2 + 50/4 + 50/8 + 50/16 + 50/32 = 25 + 12 + 6 + 3 + 1 = 47
    • Type 22: Check for Palindrome Numbers
      Example: Is 1221 a palindrome? ⇒ Yes
    • Type 23: Find the Missing Digit
      Example: 3_4 is divisible by 11. Find _.
      (3 - _ + 4) divisible by 11 ⇒ _ = 4 (as 3 - 4 + 4 = 3)
    • Type 24: Numbers Ending with a Given Digit
      Example: Unit digit of 9¹⁹ = 9
    • Type 25: Number of Digits in a Large Number
      Example: Digits in 5⁶⁰ = ⌊60×log₁₀(5)⌋ + 1 ≈ 42
    • Type 26: Find Greatest Number Dividing Multiple Values
      Example: Greatest number dividing 75, 100, 125 = HCF = 25
    • Type 27: Find Least Number Leaving Same Remainder
      Example: Least number which when divided by 4, 5, 6 leaves 3 in each case = LCM(4,5,6) + 3 = 60 + 3 = 63
    • Type 28: Numbers Formed with Digits
      Example: How many 3-digit numbers can be formed from 1,2,3,4 with no repetition?
      = 4 × 3 × 2 = 24 numbers
    • Type 29: Sum of Digits
      Example: Sum of digits of 1234 = 1 + 2 + 3 + 4 = 10
    • Type 30: Find Number between Multiples
      Example: Number between 100–200 divisible by 7 = Start from 105 → 105, 112… ⇒ Count terms in AP
    • Type 31: Perfect Cube Check
      Example: Is 512 a perfect cube? ⇒ 8³ = 512 ⇒ Yes
    • Type 32: Smallest 5-digit Number Divisible by X
      Example: Smallest 5-digit number divisible by 23 ⇒ ceil(10000/23) = 435 ⇒ 435×23 = 10005
    • Type 33: Find Smallest Number to Subtract
      Example: Smallest number to subtract from 1000 to make it divisible by 37 ⇒ 1000 ÷ 37 = 27.03 ⇒ 27×37 = 999 ⇒ Subtract 1
    • Type 34: Check if Number is Composite
      Example: 91 ⇒ divisible by 7 ⇒ Composite
    • Type 35: Highest 3-digit Number Divisible by X
      Example: Highest 3-digit number divisible by 23 ⇒ floor(999/23) = 43 ⇒ 43×23 = 989
    • Type 36: Find Middle Term of Arithmetic Progression
      Example: AP: 5, 7, 9, …, 95 ⇒ Middle term = 50 (25th term of 1st to 50th)
    • Type 37: Check for Repeating Decimal
      Example: 1/3 = 0.333… ⇒ Repeating decimal
    • Type 38: Find Smallest Number Divisible by Two Numbers
      Example: LCM of 12 and 16 = 48
    • Type 39: Difference of Squares Trick
      Example: 105² – 95² = (105 + 95)(105 – 95) = 200×10 = 2000
    • Type 40: Surds and Indices Basics
      Example: √50 = √(25×2) = 5√2

    4. Tips to Remember

    • Learn divisibility rules for 2–11.
    • LCM × HCF = Product of numbers (very useful).
    • Prime numbers only divisible by 1 and itself.
    • For remainder-type questions, modular arithmetic helps.
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