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Senior Manager  Joined: 21 Oct 2013
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Does integer n have 2 factors x & y such that 1 < x < y < n?  [#permalink]

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11 00:00

Difficulty:   65% (hard)

Question Stats: 59% (01:48) correct 41% (01:45) wrong based on 357 sessions

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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.

OE
(1): 3! < n < 4! → 6 < n < 24. Every even integer can be expressed as a multiple of 2, all even numbers in range will give us a "yes" to question.
However, since primes are in range (7, 11, 13, 17, 19, 23), that cannot be factored further than itself, we can also answer question "no."
If n = 8, it has 2 integer factors 2 and 4 that fit criteria: 1 < 2 < 4 < 8 → "yes"
If n = 7, it cannot be broken down into factors that fit criteria of question → "no"
Insufficient
(2): Any positive, odd multiple of 3 is a possible value for n. For larger multiples of 3, we can easily fit criteria, giving "yes" answer. However, since any number’s smallest multiple is itself, 3 is a possible value for n that would not fit criteria, giving "no."
If n = 15, we can factor it as 3 and 5, and we fit criteria: 1< 3 < 5 < 15 → "yes"
If n = 9, we can only factor it as 3 and 3, which does not fit criteria → "no"
Insufficient
Combined: (1) limits possible values of n to those integers between 6 and 24. (2) adds limitation that n be odd multiple of 3. Possible values for n = (9, 15 and 21)
If n = 9, we answer question "no,"
If n = 15 or 21, we can answer question "yes."
Insufficient

Hi, I want to know if we have more simple solution for this question, please.
Math Expert V
Joined: 02 Sep 2009
Posts: 64949
Does integer n have 2 factors x & y such that 1 < x < y < n?  [#permalink]

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goodyear2013 wrote:
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.

OE
(1): 3! < n < 4! → 6 < n < 24. Every even integer can be expressed as a multiple of 2, all even numbers in range will give us a "yes" to question.
However, since primes are in range (7, 11, 13, 17, 19, 23), that cannot be factored further than itself, we can also answer question "no."
If n = 8, it has 2 integer factors 2 and 4 that fit criteria: 1 < 2 < 4 < 8 → "yes"
If n = 7, it cannot be broken down into factors that fit criteria of question → "no"
Insufficient
(2): Any positive, odd multiple of 3 is a possible value for n. For larger multiples of 3, we can easily fit criteria, giving "yes" answer. However, since any number’s smallest multiple is itself, 3 is a possible value for n that would not fit criteria, giving "no."
If n = 15, we can factor it as 3 and 5, and we fit criteria: 1< 3 < 5 < 15 → "yes"
If n = 9, we can only factor it as 3 and 3, which does not fit criteria → "no"
Insufficient
Combined: (1) limits possible values of n to those integers between 6 and 24. (2) adds limitation that n be odd multiple of 3. Possible values for n = (9, 15 and 21)
If n = 9, we answer question "no,"
If n = 15 or 21, we can answer question "yes."
Insufficient

Hi, I want to know if we have more simple solution for this question, please.

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.

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Math Expert V
Joined: 02 Sep 2009
Posts: 64949
Re: Does integer n have 2 factors x & y such that 1 < x < y < n?  [#permalink]

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1
Bunuel wrote:
goodyear2013 wrote:
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.

OE
(1): 3! < n < 4! → 6 < n < 24. Every even integer can be expressed as a multiple of 2, all even numbers in range will give us a "yes" to question.
However, since primes are in range (7, 11, 13, 17, 19, 23), that cannot be factored further than itself, we can also answer question "no."
If n = 8, it has 2 integer factors 2 and 4 that fit criteria: 1 < 2 < 4 < 8 → "yes"
If n = 7, it cannot be broken down into factors that fit criteria of question → "no"
Insufficient
(2): Any positive, odd multiple of 3 is a possible value for n. For larger multiples of 3, we can easily fit criteria, giving "yes" answer. However, since any number’s smallest multiple is itself, 3 is a possible value for n that would not fit criteria, giving "no."
If n = 15, we can factor it as 3 and 5, and we fit criteria: 1< 3 < 5 < 15 → "yes"
If n = 9, we can only factor it as 3 and 3, which does not fit criteria → "no"
Insufficient
Combined: (1) limits possible values of n to those integers between 6 and 24. (2) adds limitation that n be odd multiple of 3. Possible values for n = (9, 15 and 21)
If n = 9, we answer question "no,"
If n = 15 or 21, we can answer question "yes."
Insufficient

Hi, I want to know if we have more simple solution for this question, please.

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=6, 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.

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Does integer n have 2 factors x & y such that 1 < x < y < n?  [#permalink]

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1
goodyear2013 wrote:
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.

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GRE 1: Q169 V154 Re: Does integer n have 2 factors x & y such that 1 < x < y < n?  [#permalink]

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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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Re: Does integer n have 2 factors x & y such that 1 < x < y < n?  [#permalink]

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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

VeritasPrepKarishma wrote:
goodyear2013 wrote:
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 6 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.

Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 10629
Location: Pune, India
Re: Does integer n have 2 factors x & y such that 1 < x < y < n?  [#permalink]

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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

VeritasPrepKarishma wrote:
goodyear2013 wrote:
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 6 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.

Yes, you are right. Consider only the odd multiples. n = 15 will have x and y.
Edited.
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Re: Does integer n have 2 factors x & y such that 1 < x < y < n?  [#permalink]

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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.

thanks
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Posts: 10629
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Re: Does integer n have 2 factors x & y such that 1 < x < y < n?  [#permalink]

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Mudit27021988 wrote:
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.

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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Re: Does integer n have 2 factors x & y such that 1 < x < y < n?  [#permalink]

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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.

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Re: Does integer n have 2 factors x & y such that 1 < x < y < n?  [#permalink]

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_________________ Re: Does integer n have 2 factors x & y such that 1 < x < y < n?   [#permalink] 01 Jun 2020, 05:31

# Does integer n have 2 factors x & y such that 1 < x < y < n?  