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Mathematics Dictionary
Dr. K. G. Shih
Abundant Numbers


  • Q01 | - Abundant numbers
  • Q02 | - Abundant numbers from 1 to 39
  • Q03 | - Abundant numbers from 40 to 69
  • Q04 | - Abundant numbers from 70 to 99
  • Q05 | - Abundant numbers 945


    • Q01. Abundant numbers

    Definition
    • Sum of the factors of a number n is greater then n
    • Number itself is not included
    Example
    • Factors of 12 : 1, 2, 3, 4, 6
    • Sum of factors = 1 + 2 + 3 + 4 + 6 = 16
    • Hence 12 is an abundant number

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    Q02. Abundant numbers from 1 to 39 

       Factors of 12 : 1   2   3
                       6   4   
   Sum = 1 + 2 + 3 + 4 + 6 = 16

   Factors of 18 : 1   2   3
                       9   6
   Sum = 1 + 2 + 3 + 6 + 9 = 21

   Factors of 20 : 1   2   4
                      10   5
   Sum of factors = 1 + 2 + 4 + 5 + 10 = 22

   Factors of 24 : 1   2   3   4
                      12   8   6
   Sum of factors = 1 + 2 + 3 + 4 + 6 + 8 + 12 = 36

   Factors of 30 : 1   2   3   5
                      15  10   6
   Sum of factors = 1 + 2 + 3 + 5 + 6 + 10 + 15 = 42

   Factors of 36 : 1   2   3   4   6
                      18  12   9   6
   Sum of factors = 1 + 2 + 3 + 4 + 6 +9 + 12 + 18 = 55

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    Q03. Abundant numbers from 40 to 69

    
   Factors of 40 : 1   2   4   5
                      20  10   8
   Sum of factors = 1 + 2 + 4 + 5 + 8 + 10 + 20 = 50

   Factors of 42 : 1   2   3   6
                      21  14   7
   Sum of factors = 1 + 2 + 3 + 6 + 7 + 14 + 21 = 54

   Factors of 48 : 1   2   3   4   6
                      24  16  12   8
   Sum of factors = 1 + 2 + 3 + 4 + 6 + 8 + 12 + 16 + 24 = 76

   Factos of 54 : 1   2   3   6
                     27  18   9
   Sum of factors = 1 + 2 + 3 + 6 + 9 + 18 + 27 = 66

   Factors of 56 : 1   2   4   7
                      28  14   8
   Sum of factors = 1 + 2 + 4 + 7 + 8 +14 + 28 = 64

   Factors of 60 : 1   2   3   4   5   6
                      30  20  15  12  10
   Sum of factors = 1 + 2 + 3 + 4 + 5 + 6 + 10 + 12 + 15 + 20 + 30 = 108

   Factors of  66 : 1   2   3   6
                       33  22  11
   Sum of factors = 1 + 2 + 3 + 6 + 11 + 22 + 33 = 78

   Factors of 70 : 1   2   5   7
                      35  14  10
   Sum of factors = 1 + 2 + 5 + 7 + 10 + 14 + 35 = 74

   Factors of 72 : 1   2   3   4   8      
                      36  24  18   9
   Sum of factors = 1 + 2 + 3 + 4 + 8 + 9 + 18 + 24 + 36 = 105

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    Q04. Abundant numbers from 70 to 99

    
 

   Factors of 70 : 1   2   5   7
                      35  14  10
   Sum of factors = 1 + 2 + 5 + 7 + 10 + 14 + 35 = 74

   Factors of 72 : 1   2   3   4   8      
                      36  24  18   9
   Sum of factors = 1 + 2 + 3 + 4 + 8 + 9 + 18 + 24 + 36 = 105

   Factors of 78 : 1   2   3   6
                      39  26  13
   Sum of factors = 1 + 2 + 3 + 6 + 13 + 26 + 39 = 90

   Factors of 80 : 1   2   4   5   8
                      40  20  16  10
   Sum of factors = 1 + 2 + 4 + 5 + 8 + 10 + 16 + 20 + 40 = 106

   Factors of 84 : 1   2   3   4   6   7
                      42  28  21  14  12
   Sum of factors = 1 + 2 + 3 + 4 + 6 + 7 + 12 + 14 + 21 + 28 + 42 = 130

   Factors of 88 : 1   2   4   8
                      44  22  11
   Sum of factors = 1 + 2 + 4 + 8 + 11 + 22 + 44 = 92

   Factors of 90 : 1   2   3   5   6   9
                      45  30  18  15  10
   Sum of factors = 1 + 2 + 3 + 5 + 6 + 9 + 10 + 15 + 18 + 30 + 45 =  

   Factors of 96 : 1   2   3   4   8
                      48  32  24  12
   Sum of factors = 1 + 1 + 3 + 4 + 8 + 12 + 24 + 32 + 48 = 134

   Factors of 100 : 1   2   4   5  10
                       50  25  20  10   
   Sum of factors = 1 + 2 + 4 + 5 + 10 + 20 + 25 + 50 =
    /a>
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    Q05. Abundant numbers 945

    First odd number is abundddant number 
       Factors of 945 : 1   3   5   7   9  15  21  27
                      315 189 135 105  63  45  35
   Sum of factors = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 105 + 135 + 189 + 315 = 975

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