Compare two approaches for cracking RSA key
For part(a.ii), I am presuming that "enumerating candidate prime factors" means that we have a precomputed list of all 1536-bit primes, which we can simply test one at a time.

To determine average running time we can use the prime number theorem, which states that the number of primes less that $N$ is approximately given by

$\pi(x) \sim \frac{N}{\ln N}$. 

As such, there are approximately

$\frac{2^{1536}}{\ln 2^{1536}} = 2.24 \times 10^{459}$

candidate primes to test. In reality, division of numbers this large is not an elementary operation, but even assuming it were to be one, on average we'd need half that number's worth. I think it's safe to say that (a.i) is the better choice.

As for part(b), I'm may be getting confused, but what I believe this is stating is that our key is generated by taking an input password $p$ and using it to seed a PRNG. We then generate an output stream of 1536 bits, and then select the next prime greater than our output as the first prime? And I presume this implies that we have access to said password?

If so then on average we'll have to look at

$\frac{1}{2} \cdot \ln 2^{1536} \approx 533$

different numbers before we find a prime. We'll have to do this twice though, so that's a little over a thousand primality tests on average. So the speed is then dependent on the number of elementary operations in our testing algorithm and PRNG, but assuming modern constructs (say, ISAAC and Miller-Rabin) the candidate approach should work better.

Sorry if I understand the question incorrectly. It's not the best written.