Does the integer n have two factors, x and y, such that 1 < x < y < n?

(1) 3! < n < 4! --> 6 < n < 24 --> If n=7=prime, then the answer is NO but if n=10, then the answer is YES. Not sufficient.

(2) n is odd and a multiple of 3. If n=3=prime, then the answer is NO but if n=15, then the answer is YES. Not sufficient.

(1)+(2) The same here: n=9=3^2 gives a NO answer, while n=15=3*5 gives an YES answer. Not sufficient.

Answer: E.
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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)
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Here we are asked whether n has more than two factors excluding 1 and n such that 1 < x < y < n
now statement 1 => n=> 96,24)
for 7=> NO
for 20=> YES
hence not sufficient
Statement 2 => N=3 => NO
N= 9=> No
N= 27 => YES
hence Not sufficient
Combining them N=> 9=> NO
N= 18 => YES
hence E
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Hi VeritasPrepKarishma,

Could you please clarify the highlighted part below about statement 2? I must not be understanding what the condition "n is odd" implies. I thought it could only be 3-9-15-21, but not 6-12-18, etc.

Thanks!

(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 6 it is not a prime and not a square of a prime. It has x and y.
Not sufficient

Yes, you are right. Consider only the odd multiples. n = 15 will have x and y.
Edited.
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VeritasKarishma doesnt the question imply that are there two factors x,y so that 1<x<y<n. doesnt it mean that " are there any two such factors so that 1<x<y<n". Shouldnt the ans. be yes. Say ( x=2, y =5) . such value exist?

why are we saying 1,7 wil statisfy as 1 is not less than x(x=1) in this case and it violates the condition mentioned in question.

Please clarify. too confused.

thanks

This is not given to you. It is asked.
Question: Does the integer n have two factors, x and y, such that 1 < x < y < n?
It does not give us that n has 2 factors. It asks us if it does.

(1) 3! < n < 4!
All you know here is that 6 < n < 24
We don't know that n has 2 such factors. We just know that n could be 7 or 8 or 9 etc. In some cases it will have such 2 factors, in others it won't.
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Question: Does the integer n have two factors, x and y, such that 1 < x < y < n?
Restatement: does n have minimum 4 factors?

I would start from statement 2
n may have following values.
3 (2 factors: no to main q)
9 (3 factors: no to main q)
15(4 factors: Yes to main q)
INSUFFICIENT

Statement 1: 6<n<24
9 & 15 (as already calculated) fall in between the mentioned range.
9 (3 factors: no to main q)
15 (4 factors: Yes to main q)
INSUFFICIENT

Combining:
9 & 15 all suffice statement 1&2.

9 (3 factors: no to main q)
15 (4 factors: Yes to main q)
INSUFFICIENT.

E is my answer.

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