This question is testing the number theory.
statement 1 tells about n being a prime number and nothign else. RULED OUT.
statement 2 says diff between any 2 positive factor is ODD.
Since 1 is universal factor, and for the difference to be odd the other factor should be even.
even - 0dd = odd
so it could be 2, 4, 6, 8, 10.. etc
but in case there are factors otherthan 2 thn we have factors like 1, 2, 4 or 1, 2, 6
though 2-1 = odd but 4-2 = even !
hence only exception is when it has only 2 factors. 1 & 2.
2-1 = odd. Statement 2 is sufficient