Question

What is tthe inverse of a modulo?

Answer 1

It doesn't work. It would have infinite answers.

Say 10 % 3 = 1. But 13 % 3 is also 1. And so is 16 % 3. And infinite other numbers. So the inverse modulo of 3 and 1 would yield infinite results.

Answer 2

I'm thinking of an integer ending in a 5. Can you calculate which number I'm thinking of?

No. It could be 5, or 15, or 25, or 105, and they all work equally well. You can only reverse a process if no two inputs are sent to the same output; otherwise you have no idea which to pick.

Answer 3

In modular arithmetic, we use modulo as more of a context than an operator. This problem would be phrased 10 ≅ x+2 (mod 26), which solves to x ≅ 8 (mod 26). That statement contains all the infinite answers the others were mentioning. It means that x=8+26k for some integer k, and that’s as specific as we can get.