You seem to assume that everyone knows what you are talking about--why? You certainly have not defined terms, but when I look into this, it is my impression that you have no understanding of modular arithmetic. Why not check that out?
Look, RSA is built on the fact in the private key (d,n) and the public key (e,n) d and e are exponential inverses of each other mod n. If x^d or x^e are congruent to 1 then d = phi(n) or e = phi(n) respectively... Take a look at Euler's Theorem which is a generalize form of Fermat's Little Theorem. This is why in RSA there is so much care taken in picking keys.