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Does the integer k have at least three different positive

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Re: interger K gmat Official guide [#permalink] New post 01 Sep 2009, 20:21
bhanushalinikhil wrote:
GMAT TIGER wrote:
bandit wrote:
Q. Does the integer K have at least three different positive prime factors?

1. K/15 is an integer
2. K/10 is an integer

Will post OA after some time.


C cannot be it.
It should be E because K could be 0 or 30 or any multiple of 30.



But the question is "Does the integer K has at least three different positive prime factors?". And the answer is "Yes".
0 - infinitely number of factors.
30 - 2*3*5 (3 factors).

Am I right, GT or am I missing something here? :)


Not sure.

0 is divisible by any integer/number but not sure whether all numbers are factor of 0. :oops:

As said above, the question should clearly say whether K is +ve, -ve or 0 integer.
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Re: interger K gmat Official guide [#permalink] New post 01 Sep 2009, 21:22
GMAT TIGER wrote:
0 is divisible by any integer/number but not sure whether all numbers are factor of 0. :oops:


"0 is divisible by any integer" = "any integer is a factor of 0"

That's just saying the same with different words.

All real numbers are factors of 0.

Here's the wiki entry for zero: http://en.wikipedia.org/wiki/0_(number)
And here the one for factorization, for those brave enough to go into exploring theorems: http://en.wikipedia.org/wiki/Integer_factorization
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Re: interger K gmat Official guide [#permalink] New post 01 Sep 2009, 22:06
In Gmat, only factors of +ve integers are considered factors.

powerka wrote:
GMAT TIGER wrote:
0 is divisible by any integer/number but not sure whether all numbers are factor of 0. :oops:


"0 is divisible by any integer" = "any integer is a factor of 0"

That's just saying the same with different words.

All real numbers are factors of 0.

Here's the wiki entry for zero: http://en.wikipedia.org/wiki/0_(number)
And here the one for factorization, for those brave enough to go into exploring theorems: http://en.wikipedia.org/wiki/Integer_factorization

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Re: interger K gmat Official guide [#permalink] New post 01 Sep 2009, 22:35
I've been digging a little deeper into this.

http://en.wikipedia.org/wiki/Fundamenta ... arithmetic ---> According to the Fundamental Theorem of Arithmetic any integer greater than 1 can be written as a unique product of prime numbers.

http://en.wikipedia.org/wiki/Prime_number ----> A prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself.

I dare to say that when working with primes one should assume all integers to be positive. It seems that the primality of negative numbers is not really defined. Besides, as pointed by IanStewart above, GMAT divisibility questions are always about positive integers.

Anyway, for those interested in the topic, check the following links:
http://planetmath.org/encyclopedia/Prim ... ation.html
http://www.go4expert.com/forums/showthread.php?t=18798
http://www.beatthegmat.com/negative-pri ... t1374.html

Finally, I was partially wrong in my prior post. Yes, it could be argued that if all numbers are factors of 0, all primes are factors of 0. But ... "Many authors assume 0 to be a natural number that has no prime factorization. Thus Theorem 1 of Hardy & Wright (1979) takes the form, "Every positive integer, except 1, is a product of primes,"" (taken from http://en.wikipedia.org/wiki/Fundamenta ... arithmetic) ... so, if not even the math world has decided regarding the prime factorization of 0, better to ignore it too ... GMAT would never ask about something like this.
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Re: interger K gmat Official guide [#permalink] New post 22 Sep 2009, 18:38
I'm sure the word "positive" is misplaced here. The question should read, "Does the positive integer k have at least three prime factors?" this gets rid of the option of having zero as an answer, and "positive prime" is redundant. TYPO!

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Re: interger K gmat Official guide [#permalink] New post 24 Sep 2009, 08:09
powerka wrote:
I disagree with the previous poster.
k could be 0.
This doesn't change the answer.
If k=0 then k has "at least three different positive prime factors", as every number is a factor of 0.


Totally agree with Powerka, 0=0*1*3*5*...., so 0 has at least 3 prime factors
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Re: interger K gmat Official guide [#permalink] New post 22 Nov 2009, 02:19
Not pertaining to this question but I thought the rule was

0 is a multiple for every number and a factor of none.

Is that correct? At least in GMAT
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Re: interger K gmat Official guide [#permalink] New post 22 Nov 2009, 04:27
St1: K=m15, where m is any number; Hence K can have minimum two prime factors (5,3) and let us say that m is prime than three prime factors; Insuff

St2: K=n10, where n is any number; Hence K can have minimum two prime factors (5,2) and let us say that n is prime than three prime factors; Insuff

Combined K can have minimum has to have three prime factors (5,3,2). Hence Suff

Hence C
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Re: interger K gmat Official guide [#permalink] New post 27 Nov 2009, 15:26
I agree with C.
k=2*3*5*i, so at least k has 3 different factors: 2,3,5.
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Re: interger K gmat Official guide [#permalink] New post 13 Jan 2010, 18:50
IanStewart wrote:
If this were a real GMAT question, it would ask "Does the positive integer K have at least three different positive prime factors?" GMAT questions about divisibility are always restricted to positive integers only. That said, zero is not an exception here anyway, as has been pointed out above, but you won't need to worry about that on the real test.


great point by Ian here as I was thrown off by the possibility of K being a negative integer. But now that I know GMAT questions about divisibility are only positive numbers. Thanks for the heads up!
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Re: interger K gmat Official guide [#permalink] New post 21 Aug 2010, 08:43
Hey all ,

i think we all are missing another point,

K can be negative as well

Say k=-15

therefore, k/15=-1 which is an integer,

and at the same time k can be +ve;ie k=15

hence my take would be E,
any views...
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OG 12 Diagnostic DS Error? [#permalink] New post 22 Sep 2010, 21:53
I did a search for this topic and didn't find anything, so if this is a repeat question then I apologize.

In the OG 12 diagnostic test I think I found an error in the DS section, question #42.

Does the integer k have at least three different positive prime factors?
(S1) k/15 is an integer
(S2) k/10 is an integer
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Re: OG 12 Diagnostic DS Error? [#permalink] New post 22 Sep 2010, 21:55
(S1): k is a multiple of 15. If k is 15, prime factors are 3 and 5 (i.e. <3). If k is 30, prime factors are 2,3, and 5 (i.e. =3). If k is 0, there are no prime factors. INSUFFICIENT
(S2): k is multiple of 10. If k is 10, prime factors are 2 and 5 (i.e.<3), if k is 30, once again prime factors=3. If k is 0 there are no prime factors. INSUFFICIENT
Combining S1 and S2: k must be a multiple of both 15 and 10. If k=0, there are no prime factors. If k is any other multiple of both 15 and 10 then prime factors at least include 2,3, and 5 so prime factors are > or = 3. INSUFFICIENT
Therefore the answer should be E

However, OA is C.
Can someone please explain to me where I'm going wrong or am I correct to assume that this is an error.
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Re: OG 12 Diagnostic DS Error? [#permalink] New post 22 Sep 2010, 22:14
Expert's post
cheetarah1980 wrote:
I did a search for this topic and didn't find anything, so if this is a repeat question then I apologize.

In the OG 12 diagnostic test I think I found an error in the DS section, question #42.

Does the integer k have at least three different positive prime factors?
(S1) k/15 is an integer
(S2) k/10 is an integer


cheetarah1980 wrote:
(S1): k is a multiple of 15. If k is 15, prime factors are 3 and 5 (i.e. <3). If k is 30, prime factors are 2,3, and 5 (i.e. =3). If k is 0, there are no prime factors. INSUFFICIENT
(S2): k is multiple of 10. If k is 10, prime factors are 2 and 5 (i.e.<3), if k is 30, once again prime factors=3. If k is 0 there are no prime factors. INSUFFICIENT
Combining S1 and S2: k must be a multiple of both 15 and 10. If k=0, there are no prime factors. If k is any other multiple of both 15 and 10 then prime factors at least include 2,3, and 5 so prime factors are > or = 3. INSUFFICIENT
Therefore the answer should be E

However, OA is C.
Can someone please explain to me where I'm going wrong or am I correct to assume that this is an error.



You are doing everything right till the last assumption about zero.

When we combine statements we have that 2, 3, and 5 are factors of k, so k has at least 3 different prime factors.

As for 0, question can be rephrased as follows: is k divisible by more than 3 different prime factors (is k a multiple of more than 3 prime factors). 0 is divisible by EVERY integer (except zero itself), so 0 is divisible by more than 3 prime factors.

Answer: C.

Hope it helps.
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Re: OG 12 Diagnostic DS Error? [#permalink] New post 22 Sep 2010, 22:18
Thank you so much Bunuel! I totally overlooked that property of 0. Off to update my error log.
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Re: OG 12 Diagnostic DS Error?   [#permalink] 22 Sep 2010, 22:18
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