The Egyptian 2/n table, the recto table of the Ahmes (Rhind) papyrus
The Egyptian concept of fraction requires that any fraction be represented as a sum of unit fractions without any repetitions, except 2/3 which was allowed. Thus, for example, our common fraction 2/5 would be treated as a problem, not as an answer. The problem is to divide 2 by 5; the answer would be any sum of unit fractions without repetition. One answer is 1/3 + 1/15, the preferred answer. Another possible answer would be 1/4 + 1/10 + 1/20, but that's a more complicated answer having both more terms and larger denominators. Note that 1/5 + 1/5 would not be an answer because 1/5 is repeated.
The Egyptian algorithms for mulitplication and division are based on addition, subtraction, and doubling. Therefore, one ingrediant necessary to compute products and quotients involving fractions is a table of doubles of unit fractions. It's also necessary for addition since when adding two sums of unit fractions, some particular unit fraction might occur twice.
The back (recto) of the most important Egyptian mathematical papyrus, the Ahmes, or Rhind, papyrus, includes a table of doubles of unit fractions. We can call it a 2/n table. Here it is, transcribed into modern numerals.
Note that only the denominators are listed in this transcription. In one column appears the denominator of the unit fraction to be doubled, and in the next column appear the denominators of the unit fractions for that double.
5  3 15 
7  4 28 
9  6 18 
11  6 66 
13  8 52 104 
15  10 30 
17  12 51 68 
19  12 76 114 
21  14 42 
23  12 276 
25  15 75 
27  18 54 
29  24 58 174 232 
31  20 124 155 
33  22 66 
35  30 42 
37  24 111 296 

39  26 78 
41  24 246 328 
43  42 86 129 301 
45  30 60 
47  30 141 470 
49  28 196 
51  34 102 
53  30 318 795 
55  30 330 
57  38 114 
59  36 236 531 
61  40 244 488 610 
63  42 126 
65  39 195 
67  40 355 536 
69  46 138 

71  40 568 710 
73  60 219 292 365 
75  50 150 
77  44 308 
79  60 237 316 790 
81  54 162 
83  60 332 415 498 
85  51 255 
87  58 174 
89  60 356 534 890 
91  70 130 
93  62 186 
95  60 380 570 
97  56 679 776 
99  66 198 
101  101 202 303 606 

At first glance, the only apparent regularity in the table occurs for denominators divisible by 3, and for those the rule is:
Upon further analysis, you can perceive other principles used in constructing the table.
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