RSA and prime-generator algorithms

Question

OK, my understanding of the mathematical workings of RSA may not be as deep as it should, so feel free to slap me over the head if this is stupid:

To generate a private

Answer 1

You will notice it immediatelly: the secret key will be wrong.

If p or q is composite, you choose public key (65537 typically) compute your secret key using the extended euclidean algorithm: 65537*x mod n = 1. (where n=(p-1)*(q-1))

But using the x secret key decrypt(encrypt(m)) != m In formula: (m^65537)^x mod n != m