We list below all the divisors d (2 ≤ d ≤ 102) where 10 and 100 have an inverse modulo d. An inverse is included in the table if it is a “suitably convenient” number to use in this trimming trick. For instance, 10 ^{1 }≡ 61 (mod 87), but 61 is not a particularly easy number to multiply by in mental

calculation, so it is omitted. On the other hand, 100

1

≡

20

(mod 87),

and

20 is easy to

where neither 10

1

use, so it is included. Interestingly, nor 100 ^{1 }is convenient are d = 63,

the only values for d

73, and 97.

17

5

19

2

21

2

23

7

27

8

29

3

31

3

33

10

37

11

d

10

1

100

1

3

1,

2

1,

2

7

5,

2

4,

3

1 1

3, 8,

10 9

3,

4 4 20 10 20

10

9 11 13

1

4,

1

9

10

1

100

1

4 4 30

40,

3

d

39

41

43

47

49

5

51

5

53

57

40

59

6

61

6

63

67

20

69

7

71

7

8

9 4

2 20

89

9,

80

91

9

d

73 77 79 81 83 87

93 97 99 101

10 10

10

8 8 25

1

100

10

20 8 10 40

1

1

1

Why it works: We prove the case for u ≡ 10 ^{1 }(mod d) and simply remark that the proof for v ≡ 100 ^{1 }(mod d) follows similar lines.

10