Continuity mistake: After Korwin cuts the power to the stasis pod, close-ups of the Doctor inside the pod show light still on in the pod. A shot from outside the pod, however, after McDonnell runs off to deal with Korwin, shows the inside as being dark. (00:35:25)
Continuity mistake: When the Doctor asks Captain McDonnell how many other crewmembers there are on board, he grabs her arms. After she tells him "three", the Doctor lets go of her as Scannell explains that the ship transports cargo across the galaxy. In the next shot, facing the Doctor, he's holding McDonnell's arms again. (00:03:00)
Continuity mistake: When the Doctor starts scanning Korwin, he's holding the sonic screwdriver with his thumb on the end without the light. When Ashton says that Korwin sabotaged the ship, the Doctor's grip on the screwdriver has changed to one without his thumb on the end. Shortly after, he has his thumb on the end again. (00:06:10)
Continuity mistake: When the Doctor comes up with the idea of using the generator to restart the ship's engines, in a close-up he's gesturing with his right hand, but in a wide shot is gesturing with his left. (00:10:15)
Continuity mistake: After McDonnell warns the crew that the possessed Korwin killed Abi, Ashton orders Erina to hurry up with the equipment she was sent to fetch, and she mutes the com before removing several pieces of equipment from a locker, holding them with her right hand and arm. When she closes the locker door, revealing Korwin, the equipment disappears. (00:15:20)
Character mistake: The question to unlock one of the doors asks for the next number in the sequence 313, 331, 367. The Doctor gives the answer as 379 with a convoluted (albeit correct) explanation of happy primes, but there are much simpler solutions. The difference between the first and second numbers is 18 and the difference between the second and third is 36 (which is 2 x 18), suggesting that the difference between the third number and the next could be either 54 (= 3 x 18), giving 421 as the next number, or 72 (= 2 x 36), giving 439. It's likely that the Doctor would see that the answer is ambiguous, and yet he confidently states that the most complicated answer is correct. (00:08:45 - 00:09:15)
Suggested correction: If you know what a happy prime is, like the Doctor does, it might be simpler than your idea because you gave 2 options - 421 and 439. The idea of a pattern is there should be only one answer that fits, which would be the case if they're all happy primes.
