I think I roughly understand how the RSA algorithm works.
However, I don't understand why we need the $N$, which we use as a modulus, to be $pq$ for some large primes $p, q$.
I vaguely know it has something to do with factorization, but I am kind of lost. So, hypothetical questions.
The other parts of RSA would stay the same.
RSA would still "work" with such $N$, but isn't secure for $N$ that are easily factored. If you know the factorization of $N$ (which is trivial for prime $N$s) you can calculate the private key from the public key. This totally breaks the desired security properties of RSA.
The essential equation for RSA is that $m^{\phi(N)+1}= m \mod N $ for all $m$. This works for all $N$, but only for some $N$ it's hard to calculate $\phi(N)$. When using RSA we require $\phi(N)$ being hard to calculate, since once you know $\phi(N)$ you can get $d$ from $e$ by solving $e \cdot d = 1 \mod \phi(N)$ using the extended Euclidean algorithm (just like what you do when legitimately creating the key-pair).
If $N$ has more than two factors, but at least two of those are large and hard to guess, it's still secure. But almost nobody uses this RSA variation.
