RSA: What would happen if you chose n to be a prime? If Alice were to choose n to be a prime, instead of the product of two primes, what exactly would go wrong in the RSA cryptosystem? How would Bob decrypt her message?
How would Bob decrypt her message?


That'd be easy, all we would need to assume that Bob has the public key.  If $n$ is a prime, he is able to compute $d = e^{-1} \bmod{\lambda(n)}$, because $\lambda(n) = n - 1$.  He can then raise the ciphertext to the $d$th power modulo $n$, and that's the padded plaintext.

Since anyone with the public key can do this, RSA based on a prime modulus is insecure.