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GMAT Club Legend  V
Joined: 11 Sep 2015
Posts: 4547
GMAT 1: 770 Q49 V46
Re: If the integer n is greater than 1, is n equal to 2?  [#permalink]

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dimitri92 wrote:
If the integer n is greater than 1, is n equal to 2?

(1) n has exactly two positive factors.
(2) The difference of any two distinct positive factors of n is odd.

Given: Integer n is greater than 1

Target question: Does n = 2?

Statement 1: n has exactly two positive factors.
In other words, statement 1 tells us that n is prime
There are several values of n that satisfy statement 1. Here are two:
Case a: n COULD equal 2, since 2 has exactly two positive factors: 1 and 2. In this case, the answer to the target question is YES, n equals 2
Case b: n COULD equal 3, since 3 has exactly two positive factors: 1 and 3. In this case, the answer to the target question is NO, n does NOT equal 2
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The difference of any two distinct positive factors of n is odd.
Nice!!!!
Since n is greater than 1, we know that n has at least 2 factors.
We also know that 1 is a factor of ALL positive integers, AND we know that n is also a factor of n

Some important rules:
#1. ODD +/- ODD = EVEN
#2. ODD +/- EVEN = ODD
#3. EVEN +/- ODD = ODD
#4. EVEN +/- EVEN = EVEN

Statement 2 indirectly tells us than n - 1 must be ODD
Since 1 is ODD, Rule #3 tells us that n must be EVEN
If n is EVEN, then 2 is one of the factors of n.
So far we know two of the factors of n: 1 and 2

At this point, we can conclude that 1 and 2 are the ONLY factors of n (that is, n = 2)
How can we can we conclude this?

We already know that 1 (ODD) and 2 (EVEN) are factors of n.
If there existed another factor of n, that factor would have to be EVEN or ODD
If that factor were ODD, then the difference between that number and 1 (ODD) would be EVEN, and this betrays statement 2.
If that factor were EVEN, then the difference between that number and 2 (EVEN) would be EVEN, and this betrays statement 2.
So, we can be certain that 1 and 2 are the ONLY factors of n, which means n = 2.

The answer to the target question is YES, n equals 2
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Cheers,
Brent
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Joined: 03 Aug 2017
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Re: If the integer n is greater than 1, is n equal to 2?  [#permalink]

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GMATPrepNow wrote:
dimitri92 wrote:
If the integer n is greater than 1, is n equal to 2?

(1) n has exactly two positive factors.
(2) The difference of any two distinct positive factors of n is odd.

Given: Integer n is greater than 1

Target question: Does n = 2?

Statement 1: n has exactly two positive factors.
In other words, statement 1 tells us that n is prime
There are several values of n that satisfy statement 1. Here are two:
Case a: n COULD equal 2, since 2 has exactly two positive factors: 1 and 2. In this case, the answer to the target question is YES, n equals 2
Case b: n COULD equal 3, since 3 has exactly two positive factors: 1 and 3. In this case, the answer to the target question is NO, n does NOT equal 2
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The difference of any two distinct positive factors of n is odd.
Nice!!!!
Since n is greater than 1, we know that n has at least 2 factors.
We also know that 1 is a factor of ALL positive integers, AND we know that n is also a factor of n

Some important rules:
#1. ODD +/- ODD = EVEN
#2. ODD +/- EVEN = ODD
#3. EVEN +/- ODD = ODD
#4. EVEN +/- EVEN = EVEN

Statement 2 indirectly tells us than n - 1 must be ODD
Since 1 is ODD, Rule #3 tells us that n must be EVEN
If n is EVEN, then 2 is one of the factors of n.
So far we know two of the factors of n: 1 and 2

At this point, we can conclude that 1 and 2 are the ONLY factors of n (that is, n = 2)
How can we can we conclude this?

We already know that 1 (ODD) and 2 (EVEN) are factors of n.
If there existed another factor of n, that factor would have to be EVEN or ODD
If that factor were ODD, then the difference between that number and 1 (ODD) would be EVEN, and this betrays statement 2.
If that factor were EVEN, then the difference between that number and 2 (EVEN) would be EVEN, and this betrays statement 2.
So, we can be certain that 1 and 2 are the ONLY factors of n, which means n = 2.

The answer to the target question is YES, n equals 2
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Cheers,
Brent

Hello Brent ,

Without adding the information from statement 1 aren't we keeping the possibility of N to also be a non-prime no for example

eg N can be 4 as 4 - 1 =3 ( ODD ) As both 1 and 4 are distinct factors of N ( 1, 2 and 4 ) are the distinct factors of N ie 4

also as per your solution, 2 is the value of N as 2 and 1 are the factors... we can only conclude this provided we know N is a prime number and that can only happen when we combine both statements? PLease explain where is my thought process going wrong?
GMAT Club Legend  V
Joined: 11 Sep 2015
Posts: 4547
GMAT 1: 770 Q49 V46
Re: If the integer n is greater than 1, is n equal to 2?  [#permalink]

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Top Contributor
mimajit wrote:
Hello Brent ,

Without adding the information from statement 1 aren't we keeping the possibility of N to also be a non-prime no for example

eg N can be 4 as 4 - 1 =3 ( ODD ) As both 1 and 4 are distinct factors of N ( 1, 2 and 4 ) are the distinct factors of N ie 4

also as per your solution, 2 is the value of N as 2 and 1 are the factors... we can only conclude this provided we know N is a prime number and that can only happen when we combine both statements? PLease explain where is my thought process going wrong?

Be careful; statement 2 says "The difference of any two distinct positive factors of n is odd"
So, while 4-1 is odd, 4-2 is even
So, N = 4 doesn't satisfy statement 2.
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Manager  B
Joined: 03 Aug 2017
Posts: 101
Re: If the integer n is greater than 1, is n equal to 2?  [#permalink]

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GMATPrepNow wrote:
mimajit wrote:
Hello Brent ,

Without adding the information from statement 1 aren't we keeping the possibility of N to also be a non-prime no for example

eg N can be 4 as 4 - 1 =3 ( ODD ) As both 1 and 4 are distinct factors of N ( 1, 2 and 4 ) are the distinct factors of N ie 4

also as per your solution, 2 is the value of N as 2 and 1 are the factors... we can only conclude this provided we know N is a prime number and that can only happen when we combine both statements? PLease explain where is my thought process going wrong?

Be careful; statement 2 says "The difference of any two distinct positive factors of n is odd"
So, while 4-1 is odd, 4-2 is even
So, N = 4 doesn't satisfy statement 2.

oh Ok got it ! Thanks Re: If the integer n is greater than 1, is n equal to 2?   [#permalink] 01 Nov 2019, 00:52

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