goodyear2013
Does the integer n have two factors, x and y, such that 1 < x < y < n?
(1) 3! < n < 4!
(2) n is odd and a multiple of 3.
The question is not difficult if you understand the theory of factors properly.
Does n have two factors x and y such that x and y lie between 1 and n and are distinct?
When does a number have factors between 1 and itself? When it is a composite (not a prime) number. Every composite number has a factor in between 1 and itself.
When will the factors be distinct i.e. when does the number have more than 1 factors? When it is not a perfect square or a prime number. A perfect square of a prime number such as 4 has only 1 factor between 1 and itself (1, 2, 4).
So we want two things in our n : It should not be prime and it should not be square of a prime number.
(1) 3! < n < 4!
This means 6 < n < 24
If n is 7, it is prime. It has no x and y.
If it is 8 it is not a prime and not a square of a prime. It has x and y.
Not sufficient
(2) n is odd and a multiple of 3.
If n is 3, it is prime. It has no x and y.
If it is 15, it is not a prime and not a square of a prime. It has x and y.
Not sufficient
Using both, n could be 9/12/15 etc
9 is the square of a prime. It has no x and y.
12 is not a prime and not the square of a prime. It has x and y.
Not sufficient.
Answer (E)