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## Distinct factor (help)

This topic has 2 expert replies and 6 member replies
aloha Junior | Next Rank: 30 Posts
Joined
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Posted:
15 messages

#### Distinct factor (help)

Sat Oct 11, 2008 12:58 pm
I am revising OG 11 for the second time but first time I didn't have problem with this particular DS question and now I just can't get it. I need help.
Its OG 11 DS # 132.
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 distint positive factors of n is odd.

The solution states statement B. But there are other integers >2 for example:28 is greater than 2 and the difference of any two distinct factor of 28 is odd (since 7-2=5).
Also statement 2 does not state that the 2 distinct factors of n are the only factors of n (which would be true when n=2).

### GMAT/MBA Expert

Ian Stewart GMAT Instructor
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Tue Oct 21, 2008 12:31 pm
k2gopal wrote:
Ian Stewart wrote:
Statement 1 just says "n is prime." So not sufficient.

Statement 2 is more interesting:

Suppose n is even, and n > 2. Well, if n is not equal to 2, and n is even, then 2 is a divisor of n. So n and 2 are even, and therefore n-2 is even. This contradicts what we're told in statement 2, so, if n > 2, n can't be even.
Hi Ian,

the bolded statement imho doesn't make sense. Statement 2 is contradicted ONLY if n and 2 were the only factors. The second stem as aloha rightly pointed out says "the difference of any two distinct positive factors of n is odd. " of the two factors. Given that, for the number 28, "7" and "2" are definitely distinct positive factors.

OG11 is almost never wrong, but I strongly feel that their wording should have been distinct prime factors. But again this would cause a problem because the number 1 is not prime. So I'm a little lost with this sum as well. I still feel C would be the right choice. I have no clue still how B figures.

Could anyone please re-look this question, I haven't found a convincing answer yet anywhere.
You and aloha above are both misinterpreting the second statement. When it says

"the difference of any two distinct positive factors of n is odd"

this does *not* mean:

'the difference between some pair of distinct positive factors of n is odd"

which seems to be the way you have both understood it. Instead it means

"the difference between every pair of distinct positive factors of n is odd".

That is, it means that for *any* two factors you of n that you choose, the difference will *always* be odd.

So what could n be?

* If n is even, and n > 2, then 2 and n are both distinct factors of n, and n-2 is even. So n can't be an even number greater than 2;

* If n is odd, and n > 1, then 1 and n are both distinc factors of n, and n-1 is even. So n can't be an odd number greater than 1.

Since we are told n > 1, that only leaves one possibility: n = 2.

_________________
If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com

aloha Junior | Next Rank: 30 Posts
Joined
20 Jun 2008
Posted:
15 messages
Tue Oct 21, 2008 1:06 pm
k2gopal wrote:
Ian Stewart wrote:
Statement 1 just says "n is prime." So not sufficient.

Statement 2 is more interesting:

Suppose n is even, and n > 2. Well, if n is not equal to 2, and n is even, then 2 is a divisor of n. So n and 2 are even, and therefore n-2 is even. This contradicts what we're told in statement 2, so, if n > 2, n can't be even.
Hi Ian,

the bolded statement imho doesn't make sense. Statement 2 is
contradicted ONLY if n and 2 were the only factors. The second stem as aloha rightly pointed out says "the difference of any two distinct positive factors of n is odd. " of the two factors. Given that, for the number 28, "7" and "2" are definitely distinct positive factors.

OG11 is almost never wrong, but I strongly feel that their wording should have been distinct prime factors. But again this would cause a problem because the number 1 is not prime. So I'm a little lost with this sum as well. I still feel C would be the right choice. I have no clue still how B figures.

Could anyone please re-look this question, I haven't found a convincing answer yet anywhere.
I was first very confused with this problem but if you think this way (looking at statement 2) ...Its talking about the integer (n) which has certain factors and the difference between those factors has to be odd always. So if you look at the example that I mentioned earlier ...28 the factors of which are 1,2,4,7,14,28. N should fulfill the requrement stated in statement 2 which 28 doesn't...
For example
7-4=3 (satisfies)
7-2=5 (satisfies) but
28-14=14 (doesn't satisfy). So stement 2 is not talking about 28.
According to statemnt 2 n should satisfy the requirement and anly n=2 does satisfy this not any other integer. Hope it helps.

parallel_chase Legendary Member
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Sat Oct 11, 2008 2:08 pm
aloha wrote:
2)the difference of any two distint positive factors of n is odd.
This statement actually means that difference between any two factors of n out of all the factors of n is odd

if n= 28
factors of n = 1,2,4,7,14,28
4-2 = 2 even
14-4 = 10 even
28-14 = 14 even

Therefore, if we follow the above statement n can never be 28.

it is best to try with prime integers because if we try with odd non prime integers, factors can be two odd numbers, the difference of two odd factors is even, and same goes with even non prime integers, difference of any two even integers is also even integer.

every odd prime integer will give us the difference of even number.
therefore only one situation fits the criteria i.e. if n=2, factor: 1, 2
2-1 = 1

Hope this helps.

### GMAT/MBA Expert

Ian Stewart GMAT Instructor
Joined
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Posted:
2285 messages
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Sat Oct 11, 2008 4:04 pm
aloha wrote:
I am revising OG 11 for the second time but first time I didn't have problem with this particular DS question and now I just can't get it. I need help.
Its OG 11 DS # 132.
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 distint positive factors of n is odd.

The solution states statement B. But there are other integers >2 for example:28 is greater than 2 and the difference of any two distinct factor of 28 is odd (since 7-2=5).
Also statement 2 does not state that the 2 distinct factors of n are the only factors of n (which would be true when n=2).
Statement 1 just says "n is prime." So not sufficient.

Statement 2 is more interesting:

Suppose n is even, and n > 2. Well, if n is not equal to 2, and n is even, then 2 is a divisor of n. So n and 2 are even, and therefore n-2 is even. This contradicts what we're told in statement 2, so, if n > 2, n can't be even.

Okay, suppose n is odd. Well, then 1 is a factor of n, and since n is a factor of n, n-1 must be even (since odd-odd = even). But this contradicts what we're told in Statement 2. So n can't be odd.

There's only one possibility we haven't considered yet: n = 2. All other cases are impossible, so n must be equal to 2, and statement 2 is sufficient.

_________________
If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com

aloha Junior | Next Rank: 30 Posts
Joined
20 Jun 2008
Posted:
15 messages
Sun Oct 12, 2008 1:16 pm
parallel_chase wrote:
aloha wrote:
2)the difference of any two distint positive factors of n is odd.
This statement actually means that difference between any two factors of n out of all the factors of n is odd

if n= 28
factors of n = 1,2,4,7,14,28
4-2 = 2 even
14-4 = 10 even
28-14 = 14 even

Therefore, if we follow the above statement n can never be 28.

it is best to try with prime integers because if we try with odd non prime integers, factors can be two odd numbers, the difference of two odd factors is even, and same goes with even non prime integers, difference of any two even integers is also even integer.

every odd prime integer will give us the difference of even number.
therefore only one situation fits the criteria i.e. if n=2, factor: 1, 2
2-1 = 1

Hope this helps.
Thanks a bunch for your reply. One think I want to mention...
If we look at #2 the information we have are: n>1 and the difference of any 2 positive factors of n is odd. So if we use the same example of "28" it is also true that two of the 6 factors of 28 are 7 and 2 and the difference between these two is odd. Again, difference between 7 and 4 is also odd .So n is not necessarily 2. I thought we can be sure about that only if we use both the statements. I am really sorry but am I still missing some thing?

conomav Junior | Next Rank: 30 Posts
Joined
15 Aug 2008
Posted:
14 messages
Fri Oct 17, 2008 5:39 pm
what is difference between distinct positive factor & positive factor

k2gopal Junior | Next Rank: 30 Posts
Joined
30 May 2007
Posted:
21 messages
Mon Oct 20, 2008 2:25 pm
Ian Stewart wrote:
Statement 1 just says "n is prime." So not sufficient.

Statement 2 is more interesting:

Suppose n is even, and n > 2. Well, if n is not equal to 2, and n is even, then 2 is a divisor of n. So n and 2 are even, and therefore n-2 is even. This contradicts what we're told in statement 2, so, if n > 2, n can't be even.
Hi Ian,

the bolded statement imho doesn't make sense. Statement 2 is contradicted ONLY if n and 2 were the only factors. The second stem as aloha rightly pointed out says "the difference of any two distinct positive factors of n is odd. " of the two factors. Given that, for the number 28, "7" and "2" are definitely distinct positive factors.

OG11 is almost never wrong, but I strongly feel that their wording should have been distinct prime factors. But again this would cause a problem because the number 1 is not prime. So I'm a little lost with this sum as well. I still feel C would be the right choice. I have no clue still how B figures.

Could anyone please re-look this question, I haven't found a convincing answer yet anywhere.

andyd Newbie | Next Rank: 10 Posts
Joined
31 Aug 2008
Posted:
4 messages
Mon Jun 01, 2009 4:53 pm
Nevermind - sorry for the bump. It finally made sense

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