Explaining MENSA's IQ Problem on Alien Fingers

Question

There is one probably quite well-known IQ puzzle below from Mensa:

There are a number of aliens in a room. Each alien has more than one finger on each hand. All aliens have the same number of fingers as each other. All aliens have a different number of fingers on each hand. If you knew the total number of fingers in the room you would know how many aliens were in the room. There are between $200$ and $300$ alien fingers in the room. How many aliens are in the room?

--- Puzzle courtesy of MENSA.

I know the answer is $17$, the explanation is that the number of fingers is the square of a prime, the only squares between $200$ and $300$ are $15^2$, $16^2$ and $17^2$, and only $17^2$ is the square of a prime, so there are $17$ aliens in the room.

It is a beautiful puzzle, but I still don't get two of the parts where it says:

1. "Each alien has more than one finger on each hand" - This means that the option for an alien having $1$ hand with $1$ finger and the other with $16$ fingers is not possible, thus the number of aliens will be less than $17$.
2. "All aliens have a different number of fingers on each hand" - This means that if there is one alien having, say, $6$ fingers on the left hand and $11$ on the right hand, and another alien having $11$ on the left hand and $6$ fingers on the right hand, the statement would be violated, therefore the number of aliens will be much smaller than $17$.
3. Essentially, the only possible combinations of number of fingers is $$(2,15)\; (3,14)\; (4,13)\; (5,12)\; (6,11)\; (7,10)\; (8,9)\,,$$ so $17$ is not actually a correct answer.

Is there anything I have missed or is it that the wording of the puzzle is incomplete?

Answer 1

If $n$ is the number of aliens who all have hands with a, b, c, d fingers on multiple hands) then total number of fingers is $T = n(a+b+c+d$....). As we can deduce from this the number of aliens and the answer is unique then the factorization of T must be unique into two primes. If $n$ and $(a+b+c+d...)$ are different primes this would lead to two answers for the number of aliens therefore $n = a+b+c$.... So $T=n^2$ where $n$ is prime. If $200 < T < 300$ then $n=17$.