Playing with Numbers

• Factors and Multiples

  • Factors
    • Exact divisors of a given number
    • Number of factors is finite
    • 1 is a factor of every number
    • Every number is a factor of itself
    • Factor is always less than or equal to the number
  • Multiples
    • Products obtained by multiplying a number by integers
    • Number of multiples is infinite
    • Every number is a multiple of itself
    • Multiple is always greater than or equal to the number

• Classification of Numbers

  • Prime Numbers
    • Numbers having exactly two factors (1 and the number itself)
    • Example: 2, 3, 5, 7, 11
    • 2 is the smallest and only even prime number
  • Composite Numbers
    • Numbers having more than two factors
    • Example: 4, 6, 8, 9, 10
  • Special Case (1)
    • Neither prime nor composite
  • Even and Odd Numbers
    • Even: Multiples of 2 (ends in 0, 2, 4, 6, 8)
    • Odd: Not multiples of 2 (ends in 1, 3, 5, 7, 9)
  • Co-prime Numbers
    • Two numbers having only 1 as a common factor
    • Example: (2, 3), (8, 9)
  • Twin Primes
    • Two prime numbers with a difference of 2
    • Example: (3, 5), (11, 13)

• Divisibility Tests

  • Divisibility by 2: Unit digit is 0, 2, 4, 6, or 8
  • Divisibility by 3: Sum of digits is divisible by 3
  • Divisibility by 4: Number formed by last 2 digits is divisible by 4
  • Divisibility by 5: Unit digit is 0 or 5
  • Divisibility by 6: Number is divisible by both 2 and 3
  • Divisibility by 8: Number formed by last 3 digits is divisible by 8
  • Divisibility by 9: Sum of digits is divisible by 9
  • Divisibility by 10: Unit digit is 0
  • Divisibility by 11: Difference between sum of digits at odd places and even places is 0 or divisible by 11

• Prime Factorization

  • Expressing a number as a product of prime factors only
  • Methods
    • Factor Tree Method: Branching into prime factors
    • Division Method: Continuous division by prime numbers

• Highest Common Factor (HCF)

  • Also called Greatest Common Divisor (GCD)
  • Highest of all common factors of given numbers
  • Methods to Find HCF
    • Listing factors
    • Prime factorization method
    • Division method

• Lowest Common Multiple (LCM)

  • Lowest of all common multiples of given numbers
  • Methods to Find LCM
    • Listing multiples
    • Prime factorization method
    • Common division method

• Important Properties & Applications

  • Property: Product of HCF and LCM = Product of the two numbers
  • Application of HCF: Finding maximum capacity, largest tile size, maximum length
  • Application of LCM: Finding common intervals (bells ringing together, traffic lights, steps overlapping)