Given a number, produce another random number that is the same every time and distinct from all other results

Question

Basically, I would like help designing an algorithm that takes a given number, and returns a random number that is unrelated to the first number. The stipulations being that

Answer 1

You can not have completely unrelated (particularly if you want the reverse as well).

There is a concept of modulo inverse of a number, but this would work only if the range number is a prime, eg. 100 will not work, you would need 101 (a prime). This can provide you a pseudo random number if you want.

Here is the concept of modulo inverse:

If there are two numbers a and b, such that

(a * b) % p = 1

where p is any number, then

a and b are modular inverses of each other.

For this to be true, if we have to find the modular inverse of a wrt a number p, then a and p must be co-prime, ie. gcd(a,p) = 1

So, for all numbers in a range to have modular inverses, the range bound must be a prime number.

A few outputs for range bound 101 will be:

1 == 1
2 == 51
3 == 34
4 == 76
etc.

EDIT:

Hey...actually you know, you can use the combined approach of modulo inverse and the method as defined by @Paul. Since every pair will be unique and all numbers will be covered, your random number can be:

random(k) = randomUniqueNumber(ModuloInverse(k), p)      //this is Paul's function