Playing with Numbers

Jul 9, 2024

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Lecture Notes: Playing with Numbers

Introduction

  • Welcome back, students. We will start Chapter 3: Playing with Numbers.
  • This chapter is interesting and relates to multiplication tables, factors, multiples, etc.
  • Important to remember tables up to 20 as it helps greatly.

Topics Covered

  1. Factors and Multiples
    • Definitions and Concepts
    • Examples and Real-life Applications
  2. Classification of Numbers
    • Even and Odd Numbers
    • Prime and Composite Numbers
    • Co-prime Numbers
  3. Prime Factorization
  4. LCM and HCF
  5. Divisibility Rules

Factors and Multiples

Factors

  • Definition: Numbers that divide a given number exactly (without a remainder).
  • Example: Factors of 6 are 1, 2, 3, and 6.
  • **Properties: **
    • 1 is a factor of every number.
    • Every number is a factor of itself.
    • Factors of a number are always less than or equal to that number.
    • Factors of a number are finite.

Multiples

  • Definition: Product of a number with an integer.
  • Example: Multiples of 6 are 6, 12, 18, etc.
  • Properties:
    • Multiples of a number are greater than or equal to that number.
    • Multiples are infinite.

Types of Numbers

Prime and Composite Numbers

  • Prime Numbers: Numbers with exactly 2 factors (1 and the number itself). Example: 2, 3, 5, etc.
  • Composite Numbers: Numbers with more than 2 factors. Example: 4, 6, 8, etc.
  • **Important Points: **
    • 1 is neither prime nor composite.
    • Smallest prime number is 2 (also the only even prime number).
    • All other primes are odd.

Co-prime Numbers

  • Definition: Two numbers with only 1 as their common factor. Example: 8 and 15.

Even and Odd Numbers

  • Even Numbers: Divisible by 2 (e.g., 2, 4, 6, etc.)
  • Odd Numbers: Not divisible by 2 (e.g., 1, 3, 5, etc.)

Perfect Numbers

  • Definition: A number whose sum of all factors (excluding itself) is equal to the double of that number.
  • Examples: 6, 28, 496.

Prime Factorization

  • Expressing a number as a product of primes.
  • Methods:
    • Tree Method: Example 70 тЖТ 2 ├Ч 5 ├Ч 7
    • Division Method: Repeatedly divide by prime numbers.

LCM and HCF

HCF (Highest Common Factor)

  • Definition: Highest number that is a factor of two or more numbers.
  • Methods:
    • Prime Factorization: Example, HCF of 60 and 72 is 12.
    • Division Method: Repeatedly divide by common primes.

LCM (Least Common Multiple)

  • Definition: Smallest number that is a multiple of two or more numbers.
  • Methods:
    • List Method: List multiples and find the smallest common multiple.
    • Prime Factorization: Multiply highest powers of prime factors.
  • Example: LCM of 10, 15, 20 is 60.

Divisibility Rules

  1. Divisibility by 2: Last digit is even.
  2. Divisibility by 3: Sum of digits is divisible by 3.
  3. Divisibility by 4: Last 2 digits form a number divisible by 4.
  4. Divisibility by 5: Last digit is 0 or 5.
  5. Divisibility by 6: Number is divisible by both 2 and 3.
  6. Divisibility by 7: Double the last digit, subtract from the rest; result divisible by 7.
  7. Divisibility by 8: Last 3 digits form a number divisible by 8.
  8. Divisibility by 9: Sum of digits divisible by 9.
  9. Divisibility by 10: Last digit is 0.
  10. Divisibility by 11: Alternating sum of digits is divisible by 11.

Summary

  • Understanding of factors, multiples, prime/composite numbers, co-prime numbers, HCF, LCM, and divisibility rules is crucial.
  • Solve real-life problems using these concepts.
  • Practice problems given for more clarity.

Homework: Solve problems related to HCF and LCM, and complete the match the following.

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