Class 8 Maths A Story of Numbers Notes

Class 8 Maths A Story of Numbers Notes

Chapter 3: A Story of Numbers

Overview:
This chapter explores the history and patterns of numbers, including integers, natural numbers, and their properties. It introduces concepts such as prime numbers, composite numbers, HCF, LCM, and divisibility rules in a story-based, easy-to-understand manner.

Key Concepts

  1. Natural Numbers and Whole Numbers:
    • Natural numbers: 1, 2, 3, … (used for counting)
    • Whole numbers: 0, 1, 2, 3, … (includes zero)
  2. Prime and Composite Numbers:
    • Prime number: A number greater than 1 with exactly 2 factors (e.g., 2, 3, 5, 7)
    • Composite number: A number greater than 1 with more than 2 factors (e.g., 4, 6, 9)
    • 1 is neither prime nor composite
  3. Factors and Multiples:
    • Factor: A number that divides another number completely
    • Multiple: The product of a number and any natural number
  4. Divisibility Rules (for quick calculations):
    • 2: Last digit is even
    • 3: Sum of digits divisible by 3
    • 5: Last digit is 0 or 5
    • 9: Sum of digits divisible by 9
    • 10: Last digit is 0
  5. HCF (Highest Common Factor):
    • The largest number that divides two or more numbers exactly
    • Methods: Prime factorization, division method
  6. LCM (Lowest Common Multiple):
    • The smallest number divisible by two or more numbers
    • Methods: Prime factorization, listing multiples
  7. Relationship Between HCF and LCM:
    • For two numbers aaa and bbb: HCF×LCM=a×bHCF \times LCM = a \times bHCF×LCM=a×b
  8. Interesting Number Patterns:
    • Patterns in multiples, sums, and differences
    • Observing number patterns helps in quick calculations and problem-solving

Important Points to Remember:

  • 1 is special: neither prime nor composite
  • Prime factorization is key to finding HCF and LCM
  • Divisibility rules simplify calculations for large numbers
  • HCF and LCM have a practical application in real-life problems like arranging objects, scheduling, or dividing things equally

Examples:

  1. HCF using prime factorization:
    Find HCF of 12 and 18:
    • 12 = 22⋅32^2 \cdot 322⋅3
    • 18 = 2⋅322 \cdot 3^22⋅32
    • Common factors: 21⋅31=62^1 \cdot 3^1 = 621⋅31=6
  2. LCM using prime factorization:
    Find LCM of 12 and 18:
    • Take highest powers of all prime factors: 22⋅32=362^2 \cdot 3^2 = 3622⋅32=36
  3. Divisibility check:
    Is 324 divisible by 9?
    • Sum of digits = 3 + 2 + 4 = 9 → divisible by 9

Questions

A. Very Short Answer Questions (1–10)

  1. Write the first 5 natural numbers.
  2. Write the first 5 whole numbers.
  3. Is 1 a prime number, composite number, or neither?
  4. Find all factors of 12.
  5. Find all multiples of 7 less than 50.
  6. Check if 45 is divisible by 3.
  7. Check if 130 is divisible by 5.
  8. Find the prime factors of 18.
  9. Find the HCF of 8 and 12 using listing method.
  10. Find the LCM of 6 and 8 using listing multiples.

B. Short Answer Questions (11–25)

  1. Write the first 5 composite numbers.
  2. Check whether 97 is prime or composite.
  3. Find HCF of 24 and 36 using prime factorization.
  4. Find LCM of 15 and 20 using prime factorization.
  5. Verify that HCF×LCM=15×20HCF \times LCM = 15 \times 20HCF×LCM=15×20 for the numbers 15 and 20.
  6. Find the HCF of 48, 60, and 72.
  7. Find the LCM of 12, 15, and 20.
  8. Is 91 divisible by 7?
  9. Express 180 as a product of prime factors.
  10. Express 210 as a product of prime factors.
  11. Find the HCF of 36 and 84.
  12. Find the LCM of 18 and 24.
  13. Check if 123 is divisible by 3.
  14. Check if 245 is divisible by 5.
  15. Find HCF and LCM of 9 and 12 using prime factorization.

C. Word Problems / Application Questions (26–40)

  1. Two ropes are 24 m and 36 m long. Find the greatest length of pieces they can be cut into so that both lengths are divisible.
  2. Two bells ring at intervals of 12 minutes and 15 minutes. If they ring together at 9:00 am, when will they ring together next?
  3. A shopkeeper wants to arrange 48 pencils and 60 erasers in equal rows without any leftover. Find the number of items in each row.
  4. Find LCM of 14 and 18 and explain what it represents in a practical situation.
  5. Three numbers are 8, 12, and 20. Find their HCF and explain the meaning.
  6. The sum of two numbers is 45. Their HCF is 9. Find the numbers if one of them is 27.
  7. Find the smallest number divisible by 6, 8, and 12.
  8. A number is divisible by 2, 3, and 5. Find its smallest possible value.
  9. A factory produces pens in packs of 12 and pencils in packs of 18. What is the minimum number of packs so that both items finish together?
  10. A bus leaves a station every 15 minutes and another bus every 20 minutes. If both leave together at 9:00 am, when will they leave together next?
  11. Find HCF of 36, 48, and 60.
  12. Find LCM of 8, 9, and 12.
  13. A gardener wants to plant 48 rose bushes and 72 tulip plants in equal rows without leftovers. How many rows can he make?
  14. Find LCM of 7, 14, and 21 and explain what it represents.
  15. Three friends decide to meet after a regular interval of 9, 12, and 15 days. When will they meet next if they met today?

D. Higher Order / Thinking Questions (41–50)

  1. Find all prime numbers between 50 and 70.
  2. Write 3 numbers that are divisible by both 2 and 3.
  3. Find HCF and LCM of 56 and 98.
  4. If two numbers are 24 and 36, verify that HCF × LCM = 24 × 36.
  5. Check whether 1001 is divisible by 7.
  6. Find the smallest 3-digit number divisible by 2, 3, and 5.
  7. If HCF of two numbers is 12 and LCM is 180, and one number is 36, find the other number.
  8. A theatre has 120 chairs and 150 tables. Find the largest possible number of rows with equal chairs and tables in each row.
  9. Check divisibility of 245 by 7, 5, and 3.
  10. Write all factors of 84 and identify which are prime.

Answers

A. Very Short Answer Questions (1–10)

  1. First 5 natural numbers: 1, 2, 3, 4, 5
  2. First 5 whole numbers: 0, 1, 2, 3, 4
  3. 1 is neither prime nor composite
  4. Factors of 12: 1, 2, 3, 4, 6, 12
  5. Multiples of 7 < 50: 7, 14, 21, 28, 35, 42, 49
  6. Sum of digits of 45 = 4 + 5 = 9 → divisible by 3 → Yes
  7. Last digit of 130 = 0 → divisible by 5 → Yes
  8. Prime factors of 18: 2 × 3 × 3 or 2 × 3²
  9. HCF of 8 and 12 by listing factors:
    • 8: 1, 2, 4, 8
    • 12: 1, 2, 3, 4, 6, 12
    • Common factors: 1, 2, 4 → HCF = 4
  10. LCM of 6 and 8 by listing multiples:
    • Multiples of 6: 6, 12, 18, 24, 30…
    • Multiples of 8: 8, 16, 24, 32…
    • Smallest common multiple = 24

B. Short Answer Questions (11–25)

  1. First 5 composite numbers: 4, 6, 8, 9, 10
  2. 97 is prime (only divisible by 1 and 97)
  3. HCF of 24 and 36 using prime factorization:
    • 24 = 2³ × 3
    • 36 = 2² × 3²
    • Common factors: 2² × 3 = 12
  4. LCM of 15 and 20:
    • 15 = 3 × 5
    • 20 = 2² × 5
    • LCM = 2² × 3 × 5 = 60
  5. Verify: HCF × LCM = 15 × 20 → 5 × 60 = 300; 15 × 20 = 300 ✅
  6. HCF of 48, 60, 72:
    • 48 = 2⁴ × 3
    • 60 = 2² × 3 × 5
    • 72 = 2³ × 3²
    • Common factors: 2² × 3 = 12
  7. LCM of 12, 15, 20:
    • 12 = 2² × 3, 15 = 3 × 5, 20 = 2² × 5
    • Take highest powers: 2² × 3 × 5 = 60
  8. 91 ÷ 7 = 13 → divisible → Yes
  9. 180 = 2² × 3² × 5 (prime factorization)
  10. 210 = 2 × 3 × 5 × 7 (prime factorization)
  11. HCF of 36 and 84:
    • 36 = 2² × 3², 84 = 2² × 3 × 7 → HCF = 2² × 3 = 12
  12. LCM of 18 and 24:
    • 18 = 2 × 3², 24 = 2³ × 3 → LCM = 2³ × 3² = 72
  13. 123 sum of digits = 1 + 2 + 3 = 6 → divisible by 3 → Yes
  14. 245 last digit = 5 → divisible by 5 → Yes
  15. HCF of 9 and 12: 3; LCM of 9 and 12: 36

C. Word Problems / Applications (26–40)

  1. Ropes 24 m and 36 m → greatest length divisible by both = HCF(24,36) = 12 m
  2. Bells interval 12 min and 15 min → LCM(12,15) = 60 → next together at 10:00 am
  3. Arrange 48 pencils and 60 erasers → HCF(48,60) = 12 → 12 items per row
  4. LCM of 14 and 18 = 126 → represents first time both events coincide
  5. HCF of 8, 12, 20:
    • 8=2³, 12=2²×3, 20=2²×5 → HCF=2²=4
  6. Sum = 45, HCF = 9, one number = 27 → other = 45−27=18 ✅
  7. Smallest number divisible by 6,8,12 = LCM(6,8,12)=24
  8. Number divisible by 2,3,5 = LCM(2,3,5)=30 → smallest number = 30
  9. Pens 12, pencils 18 → LCM(12,18)=36 packs
  10. Bus intervals 15 and 20 min → LCM(15,20)=60 → next together at 10:00 am
  11. HCF(36,48,60) = 12
  12. LCM(8,9,12) = 72
  13. Rose 48, Tulip 72 → HCF(48,72)=24 rows
  14. LCM(7,14,21) = 42 → represents first multiple divisible by all three
  15. Friends meet after 9,12,15 days → LCM(9,12,15)=180 days

D. Higher Order / Thinking (41–50)

  1. Prime numbers between 50–70: 53, 59, 61, 67
  2. Numbers divisible by 2 and 3: 6, 12, 18
  3. HCF and LCM of 56 and 98:
    • 56 = 2³×7, 98=2×7² → HCF=2×7=14, LCM=2³×7²=196
  4. Verify HCF×LCM = 24×36 → 12×72=864 ✅
  5. 1001 ÷ 7 = 143 → divisible → Yes
  6. Smallest 3-digit divisible by 2,3,5 → LCM(2,3,5)=30 → 100÷30=3 remainder 10 → next multiple = 30×4=120
  7. HCF=12, LCM=180, one number=36 → other number = (HCF×LCM)/36 = (12×180)/36=60
  8. Chairs 120, Tables 150 → HCF=30 → 30 rows
  9. 245 divisible by 7? 245÷7=35 ✅, divisible by 5? last digit 5 ✅, divisible by 3? sum=11 → no ❌
  10. Factors of 84: 1,2,3,4,6,7,12,14,21,28,42,84 → prime: 2,3,7