Premise:

In modular computing each number relatively prime with the module M is a divisor of all numbers in the module range.

In Mod 90, there are 24 numbers, relatively prime with 90:

1, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89

We note that couples are complementary to 90: 1 + 89 = 90, 7 + 83 = 90, ..., 43 + 47 = 90

We can identify 12 pairs whose product a*b ? 1 mod 90:

Solve [{Mod [a * b, 90] == 1, 0 ? {a, b} ? 90}, {a, b}, Integers]

{a -> 01, b -> 01}, -> 1*1 = 1 ? 1 Mod 90

{a -> 07, b -> 13}, -> 7*13 = 91 ? 1 Mod 90

{a -> 11, b -> 41}, -> 11*41 = 451 ? 1 Mod 90

{a -> 13, b -> 07}, -> 13*7 = 91 ? 1 Mod 90

{a -> 17, b -> 53}, -> 17*53 = 901 ? 1 Mod 90

{a -> 19, b -> 19}, -> 19*19 = 361 ? 1 Mod 90

{a -> 23, b -> 47}, -> 23*47 = 1081 ? 1 Mod 90

{a -> 29, b -> 59}, -> 29*59 = 1711 ? 1 Mod 90

{a -> 31, b -> 61}, -> 31*61 = 1891 ? 1 Mod 90

{a -> 37, b -> 73}, -> 37*73 = 2701 ? 1 Mod 90

{a -> 41, b -> 11}, -> 41*11 = 451 ? 1 Mod 90

{a -> 43, b -> 67}, -> 43*67 = 2881 ? 1 Mod 90

{a -> 47, b -> 23}, -> 47*23 = 1081 ? 1 Mod 90

{a -> 49, b -> 79}, -> 49*79 = 3871 ? 1 Mod 90

{a -> 53, b -> 17}, -> 53*17 = 901 ? 1 Mod 90

{a -> 59, b -> 29}, -> 59*29 = 1711 ? 1 Mod 90

{a -> 61, b -> 31}, -> 61*31 = 1891 ? 1 Mod 90

{a -> 67, b -> 43}, -> 67*43 = 2881 ? 1 Mod 90

{a -> 71, b -> 71}, -> 71*71 = 5041 ? 1 Mod 90

{a -> 73, b -> 37}, -> 73*37 = 2701 ? 1 Mod 90

{a -> 77, b -> 83}, -> 77*83 = 6391 ? 1 Mod 90

{a -> 79, b -> 49}, -> 79*49 = 3871 ? 1 Mod 90

{a -> 83, b -> 77}, -> 83*77 = 6391 ? 1 Mod 90

{a -> 89, b -> 89}, -> 89*89 = 7921 ? 1 Mod 90

Then multiplying any number N * 1, 91, 451, 91, 901, 361, 1081, 1711, 1891, 2701, 451, 2881, 1081, 3871, 901, 1711, 1891, 2881, 5041, 2701, 6391, 3871, 6391, 7921 Mod 90 we get the same start number N.

The number obtained for example by: N * 91 = N Mod 90 = N * 7 * 13 which will be divisible both by 7 that from 13, then if you want to divide by 7 or by 13:

N / 7 = N * 7 * 13 / 7 = N * 13 Mod 90

N / 13 = N * 7 * 13 / 13 = N * 7 Mod 90

Question:

Now, if I get fractions as output, which in the modular calculation, as said above, can become integers …

{3/5, 2, 13, 19, 43/2, 295/11, 139/5, 41, 251/5, 53, 63, 191/3, 75, 997/13, 78, 82, 83, 259/3, 1145/13, 971/11, 443/5, 89}

295 / 11 = 295 * 41 = 35 mod 90

997 / 13 = 997 * 7 = 49 mod 90

1145 / 13 = 1145 * 7 = 5 mod 90

971 / 11 = 971 * 41 = 31 mod 90

If fraction is N/a how can I set a formula, based on the denominator a, that multiply the numerator N * b where a and b are a pair whose product ? 1 Mod 90? (Eg. 23/7 = 23*13 Mod 90)

thanks