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Divisibility
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Divisibility

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  • 1. The Division Algorithm
  • 2. Before we study divisibility, we must remember the division algorithm.
    r
    dividend = (divisor ⋅ quotient) + remainder
  • 3. A number is divisible by another number if the remainder is 0 and quotient is a natural number.
    Divisibility
  • 4. If a number is divided by itself then quotient is 1.
    If a number is divided by 1 then quotient is itself.
    If 0 is divided by any none zero number then quotient is 0.
    If any number is divided by zero then quotient is undefined.
    Some remarks:
  • 5. Divisibility by 2:
    A natural number is divisible by 2 if it is even, i.e. if its units (last) digit is 0, 2, 4, 6, or 8.
    Divisibility Rules
    Example: Check if each number is divisible by 2.
    a. 108 b. 466 c. 87 682 d. 68 241
    e. 76 543 010
  • 6. Divisibility by 3:
    A natural number is divisible by 3 if the sum of the digits in the number is multiple of 3.
    Divisibility Rules
    Example: Determine whether the following numbers are divisible by 3 or not.
    7605
    42 145
    c) 555 555 555 555 555
  • 7. Divisibility by 4:
    A natural number is divisible by 4 if the last two digits of the number are 00 or a multiple of 4.
    Divisibility Rules
    Example: Determine whether the following numbers are divisible by 4 or not.
    7600
    47 116
    c) 985674362549093
  • 8. Divisibility Rules
    Example: 5m3 is a three-digit number where m is a digit. If 5m3 is divisible by 3, find all the possible values of m.
    Example: a381b is a five-digit number where a and b are digits. If a381b is divisible by 3, find the possible values of a + b.
  • 9. Divisibility Rules
    Example: t is a digit. Find all the possible values of t if:
    a) 187t6 is divisible by 4.
    b) 2741t is divisible by 4.
  • 10. Divisibility Rules
    Divisibility by 5:
    A natural number is divisible by 5 if its last digit is 0 or 5.
    Example: m235m is a five-digit number where m is a digit. If m235m is divisible by 5, find all the possible values of m.
  • 11. Divisibility Rules
    Divisibility by 6:
    A natural number is divisible by 6 if it is divisible by both 2 and 3.
    Example: Determine whether the following numbers are divisible by 6 or not.
    4608
    6 9030
    c) 22222222222
  • 12. Divisibility Rules
    Example: 235mn is a five-digit number where m and n are digits. If 235mn is divisible by 5 and 6, find all the possible pairs of m, n.
  • 13. Divisibility Rules
    Divisibility by 8:
    A natural number is divisible by 8 if the number formed by last three digits is divisible by 8.
    Example: Determine whether the following number is divisible by 8 or not.
    5 793 128
    7265384
    456556
  • 14. Divisibility Rules
    Divisibility by 9:
    A natural number is divisible by 9 if the sum of the digits of the number is divisible by 9.
    Example: 365m72 is a six-digit number where m is a digit. If 365m72 is divisible by 9, find all the possible values of m.
    Example: 5m432n is a six-digit number where m and n are digits. If 5m432n is divisible by 9, find all the possible values of m + n.
  • 15. Divisibility Rules
    Divisibility by 10:
    A natural number is divisible by 10 if its units (last) digit is 0.
    Example: is 3700 divisible by 10?
  • 16. Divisibility Rules
    Divisibility by 11:
    A natural number is divisible by 11 if the difference between the sum of the odd-numbered digits and the sum of the even-numbered digits is a multiple of 11.
    Example: is 5 764 359 106 divisible by 11?