Class 6 Notes

Class 6 Mathematics Notes Chapter 3 (Playing with Numbers) – Mathematics Book

Philoid

Philoid

— 7 min read

Mathematics
Alright class, let's get focused on Chapter 3, "Playing with Numbers." This chapter is fundamental, not just for your Class 6 understanding, but it lays the groundwork for many concepts you'll encounter in quantitative aptitude sections of government exams. Mastering these basics – factors, multiples, primes, HCF, LCM, and divisibility rules – is crucial for speed and accuracy later on.

Here are the detailed notes covering the key concepts:

Chapter 3: Playing with Numbers - Detailed Notes

1. Factors and Multiples

  • Factor: A factor of a number is an exact divisor of that number.
    • Example: Factors of 12 are 1, 2, 3, 4, 6, and 12 (because 12 ÷ 1 = 12, 12 ÷ 2 = 6, 12 ÷ 3 = 4, 12 ÷ 4 = 3, 12 ÷ 6 = 2, 12 ÷ 12 = 1).
    • Properties of Factors:
      • 1 is a factor of every number.
      • Every number is a factor of itself.
      • Every factor of a number is less than or equal to the number.
      • The number of factors of a given number is finite.
  • Multiple: A multiple of a number is obtained by multiplying that number by any natural number (1, 2, 3, ...).
    • Example: Multiples of 5 are 5 (5x1), 10 (5x2), 15 (5x3), 20 (5x4), and so on.
    • Properties of Multiples:
      • Every number is a multiple of itself.
      • Every multiple of a number is greater than or equal to that number.
      • The number of multiples of a given number is infinite.

2. Prime and Composite Numbers

  • Prime Number: A natural number greater than 1 that has exactly two distinct factors: 1 and the number itself.
    • Examples: 2, 3, 5, 7, 11, 13, 17, 19...
    • Key Points:
      • 2 is the smallest prime number.
      • 2 is the only even prime number.
  • Composite Number: A natural number greater than 1 that has more than two factors.
    • Examples: 4 (factors: 1, 2, 4), 6 (factors: 1, 2, 3, 6), 8, 9, 10, 12...
  • The Number 1: The number 1 is neither prime nor composite. It has only one factor, which is 1 itself.

3. Tests for Divisibility of Numbers

These rules are extremely important for quick calculations in exams.

  • Divisibility by 2: A number is divisible by 2 if its last digit (unit's digit) is 0, 2, 4, 6, or 8 (i.e., an even number).
    • Example: 508 is divisible by 2. 129 is not.
  • Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
    • Example: 123 (1+2+3=6, 6 is divisible by 3). 568 (5+6+8=19, 19 is not divisible by 3).
  • Divisibility by 4: A number is divisible by 4 if the number formed by its last two digits (tens and units) is divisible by 4.
    • Example: 7924 (24 is divisible by 4). 813 (13 is not divisible by 4). Also, numbers ending in 00 are divisible by 4.
  • Divisibility by 5: A number is divisible by 5 if its last digit is 0 or 5.
    • Example: 195, 200.
  • Divisibility by 6: A number is divisible by 6 if it is divisible by both 2 and 3.
    • Example: 132 (It's even, so divisible by 2. Sum of digits = 1+3+2=6, which is divisible by 3. So, 132 is divisible by 6). 140 (Divisible by 2, but 1+4+0=5, not divisible by 3. So, 140 is not divisible by 6).
  • Divisibility by 8: A number is divisible by 8 if the number formed by its last three digits (hundreds, tens, units) is divisible by 8.
    • Example: 7120 (120 ÷ 8 = 15. So, 7120 is divisible by 8). 5123 (123 is not divisible by 8). Also, numbers ending in 000 are divisible by 8.
  • Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
    • Example: 729 (7+2+9=18, 18 is divisible by 9). 1234 (1+2+3+4=10, 10 is not divisible by 9).
  • Divisibility by 10: A number is divisible by 10 if its last digit is 0.
    • Example: 560.
  • Divisibility by 11: A number is divisible by 11 if the difference between the sum of the digits at odd places (from the right) and the sum of the digits at even places (from the right) is either 0 or divisible by 11.
    • Example: 61809
      • Sum of digits at odd places (9, 8, 6) = 9 + 8 + 6 = 23
      • Sum of digits at even places (0, 1) = 0 + 1 = 1
      • Difference = 23 - 1 = 22. Since 22 is divisible by 11, the number 61809 is divisible by 11.
    • Example: 13574
      • Odd places (4, 5, 1) = 4 + 5 + 1 = 10
      • Even places (7, 3) = 7 + 3 = 10
      • Difference = 10 - 10 = 0. Since the difference is 0, the number 13574 is divisible by 11.

4. Common Factors and Common Multiples

  • Common Factors: Factors that are shared by two or more numbers.
    • Example: Factors of 12 are 1, 2, 3, 4, 6, 12. Factors of 18 are 1, 2, 3, 6, 9, 18. Common factors are 1, 2, 3, 6.
  • Common Multiples: Multiples that are shared by two or more numbers.
    • Example: Multiples of 3 are 3, 6, 9, 12, 15, 18... Multiples of 4 are 4, 8, 12, 16, 20... Common multiples are 12, 24, 36...

5. Prime Factorization

  • Expressing a given number as a product of its prime factors.
  • Every composite number can be factorized into primes, and this factorization is unique (apart from the order in which the prime factors occur). This is known as the Fundamental Theorem of Arithmetic.
  • Method: Use repeated division by prime numbers starting from the smallest (2, 3, 5...).
    • Example: Prime factorization of 36:
      • 36 ÷ 2 = 18
      • 18 ÷ 2 = 9
      • 9 ÷ 3 = 3
      • 3 ÷ 3 = 1
      • So, 36 = 2 × 2 × 3 × 3 = 2² × 3²

6. Highest Common Factor (HCF)

  • Also known as Greatest Common Divisor (GCD).
  • The HCF of two or more given numbers is the highest (or greatest) of their common factors.
  • Methods to find HCF:
    • Listing Common Factors: List all factors of each number, find the common factors, and pick the largest one. (Suitable for small numbers).
    • Prime Factorization Method:
      1. Find the prime factorization of each number.
      2. Identify the common prime factors.
      3. Multiply the common prime factors, taking the lowest power of each common factor.
      • Example: Find HCF of 12 and 18.
        • 12 = 2 × 2 × 3 = 2² × 3¹
        • 18 = 2 × 3 × 3 = 2¹ × 3²
        • Common prime factors are 2 and 3.
        • Lowest power of 2 is 2¹. Lowest power of 3 is 3¹.
        • HCF = 2¹ × 3¹ = 2 × 3 = 6.
  • Co-prime Numbers: Two numbers having only 1 as a common factor are called co-prime numbers. Their HCF is 1. (e.g., 4 and 9, 15 and 16). Note: Co-prime numbers need not be prime themselves.

7. Lowest Common Multiple (LCM)

  • The LCM of two or more given numbers is the lowest (or smallest) of their common multiples.
  • Methods to find LCM:
    • Listing Common Multiples: List multiples of each number until you find the first common one. (Suitable for small numbers).
    • Prime Factorization Method:
      1. Find the prime factorization of each number.
      2. Multiply all distinct prime factors, taking the highest power of each prime factor that appears in any factorization.
      • Example: Find LCM of 12 and 18.
        • 12 = 2² × 3¹
        • 18 = 2¹ × 3²
        • Distinct prime factors are 2 and 3.
        • Highest power of 2 is 2². Highest power of 3 is 3².
        • LCM = 2² × 3² = 4 × 9 = 36.
    • Common Division Method:
      1. Arrange the numbers in a row.
      2. Divide by a prime number that divides at least one of the numbers. Carry forward numbers not divisible.
      3. Repeat until the quotients are all 1.
      4. LCM is the product of all the prime divisors.
      • Example: LCM of 12 and 18.
          2 | 12, 18
  2 |  6,  9
  3 |  3,  9
  3 |  1,  3
    |  1,  1
        • LCM = 2 × 2 × 3 × 3 = 36.

8. Relationship between HCF and LCM

  • For any two positive integers 'a' and 'b':
    Product of the two numbers = Product of their HCF and LCM
    a × b = HCF(a, b) × LCM(a, b)
  • This relationship is extremely useful for problem-solving. If you know any three of these values (the two numbers, HCF, LCM), you can find the fourth.
  • Note: This formula generally applies only to two numbers.

Multiple Choice Questions (MCQs)

Here are 10 MCQs based on the concepts discussed:

  1. Which of the following is the smallest prime number?
    a) 0
    b) 1
    c) 2
    d) 3

  2. The number 1 is:
    a) A prime number
    b) A composite number
    c) Neither prime nor composite
    d) An even number

  3. Which of the following numbers is divisible by 3?
    a) 134
    b) 567
    c) 890
    d) 211

  4. The HCF of 15 and 25 is:
    a) 1
    b) 3
    c) 5
    d) 75

  5. The LCM of 8 and 12 is:
    a) 4
    b) 8
    c) 12
    d) 24

  6. Which of the following pairs is co-prime?
    a) (6, 8)
    b) (9, 12)
    c) (7, 14)
    d) (5, 9)

  7. The prime factorization of 60 is:
    a) 2 × 3 × 10
    b) 4 × 3 × 5
    c) 2 × 2 × 3 × 5
    d) 6 × 10

  8. A number is divisible by 6 if it is divisible by:
    a) 2 and 4
    b) 2 and 3
    c) 3 and 4
    d) 2 only

  9. The product of two numbers is 120. If their HCF is 6, what is their LCM?
    a) 10
    b) 20
    c) 30
    d) 40

  10. Which number is divisible by 11?
    a) 12345
    b) 70169
    c) 98765
    d) 11223

Answer Key for MCQs:

  1. c) 2
  2. c) Neither prime nor composite
  3. b) 567 (Sum of digits 5+6+7=18, which is divisible by 3)
  4. c) 5 (Factors of 15: 1, 3, 5, 15; Factors of 25: 1, 5, 25. Highest common factor is 5)
  5. d) 24 (Multiples of 8: 8, 16, 24, 32...; Multiples of 12: 12, 24, 36... Lowest common multiple is 24)
  6. d) (5, 9) (HCF is 1)
  7. c) 2 × 2 × 3 × 5 (2² × 3 × 5)
  8. b) 2 and 3
  9. b) 20 (LCM = Product / HCF = 120 / 6 = 20)
  10. b) 70169 (Odd places: 9+1+7=17; Even places: 6+0=6; Difference = 17-6=11, which is divisible by 11)

Make sure you understand the 'why' behind each answer, especially for the divisibility rules and HCF/LCM calculations. Keep practicing!

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