Set A has 18 - 2 = 16 elements, while B has 48 - 2 = 46 elements.

The roots are all located on a unit circle in the Argand plane. The LCM of  18 and 48 is 144, so all products will be at multiples of 1/144 of a full circle (adding angles to do multiplication). Elements of A are multiples of 8/144 of a circle, and of B are multiples of 3/144 of a circle.

This program does the addition of the 144ths and computes the resulting 144ths. 0 and 72 mod 144 correspond to real values and are not used as multiplicands (addends in the additions) or products (sums in the additions).

DefDbl A-Z

Dim member(143), crlf$

Private Sub Form_Load()

 Text1.Text = ""

 crlf$ = Chr(13) + Chr(10)

 Form1.Visible = True

  

  ' count 1/144's of a circle in the Argand plane

  For a = 8 To 144 - 8 Step 8

   If a * 2 <> 144 Then

    For b = 3 To 144 - 3 Step 3

      If b * 2 <> 144 Then

        prodAngle = (a + b) Mod 144

        If prodAngle <> 0 And prodAngle <> 72 Then

          member(prodAngle) = member(prodAngle) + 1

        End If

      End If

    Next

   End If

  Next a

  

  For m = 1 To 143

    If member(m) > 0 Then

      ct = ct + 1

      Text1.Text = Text1.Text & m & Str(member(m)) & crlf

    End If

  Next

  Text1.Text = Text1.Text & ct & crlf

  Text1.Text = Text1.Text & "done"

  DoEvents

End Sub

Listed first are counts of the various possible multiples of 1/144 of a circle in the products angles on the Argand plane:

1 6

2 6

3 4

4 6

5 6

6 4

7 6

8 4

9 4

10 6

11 6

12 4

13 6

14 6

15 4

16 4

17 6

18 4

19 6

20 6

21 4

22 6

23 6

24 2

25 6

26 6

27 4

28 6

29 6

30 4

31 6

32 4

33 4

34 6

35 6

36 4

37 6

38 6

39 4

40 4

41 6

42 4

43 6

44 6

45 4

46 6

47 6

48 2

49 6

50 6

51 4

52 6

53 6

54 4

55 6

56 4

57 4

58 6

59 6

60 4

61 6

62 6

63 4

64 4

65 6

66 4

67 6

68 6

69 4

70 6

71 6

73 6

74 6

75 4

76 6

77 6

78 4

79 6

80 4

81 4

82 6

83 6

84 4

85 6

86 6

87 4

88 4

89 6

90 4

91 6

92 6

93 4

94 6

95 6

96 2

97 6

98 6

99 4

100 6

101 6

102 4

103 6

104 4

105 4

106 6

107 6

108 4

109 6

110 6

111 4

112 4

113 6

114 4

115 6

116 6

117 4

118 6

119 6

120 2

121 6

122 6

123 4

124 6

125 6

126 4

127 6

128 4

129 4

130 6

131 6

132 4

133 6

134 6

135 4

136 4

137 6

138 4

139 6

140 6

141 4

142 6

143 6

In total 142 possibilities are represented. Only the two real values are discounted from all 144 possibilities.

Of course, two of the products are purely imaginary also. Do they count as complex?