Data Sufficiency

Hi

I have problems with three questions. This is the first of three related posts. I would appreciate your help in any! Thanks.

I'm having some difficulties understanding Q132 of the Official Guide 11 in Data Sufficiency (page 331).
Question goes something like this:
132. If the integer n is greater than 1, is n equal to 2?
(i) n has exactly two positive factors
(ii) The difference of any two distinct positive factors of n is odd
The correct answer is B [statement (ii) alone is suficient]

statement (i) is not a problem. I get it why it is insufficient.
statement (ii), however, is said to be sufficient because
- if n>2 and ODD, then the difference must be EVEN.
- if n>2 and EVEN, then the difference must be EVEN. Therefore we know that statement (ii) is wrong and the statement is sufficient.


I agree that if n is odd, then the diff is even. but I don't agree with the statement when n is EVEN.
For example, n could be 10, in which case the diff between factors 2 and 5 is 3, which is ODD.
n could also be 12, in which case the diff btw the factors 6 and 2, is 4, which is EVEN.

I therefore think that statement (ii) is insufficient, but apparently OG11 disagrees. Can someone expain the OG reasoning please?

Thanks!

Check this link

https://www.beatthegmat.com/factors-t63310.html

when n=10, it has 1,2,5,10 as factors.

Ok, got it.
I was misinterpreting the question to mean that if any *one* difference between a pair of factors was odd, then statement (ii) was true. But if all the factor pairs must be odd, I see why we are limited to n=2.
Thanks.