Author 
Message 
TAGS:

Hide Tags

Manager
Joined: 04 Apr 2010
Posts: 162

If k is a positive integer, is k a prime number? [#permalink]
Show Tags
19 Jun 2010, 11:37
3
This post received KUDOS
16
This post was BOOKMARKED
Question Stats:
45% (02:22) correct
55% (01:29) wrong based on 391 sessions
HideShow timer Statistics
If k is a positive integer, is k a prime number? (1) No integers between 2 and \(\sqrt{k}\), inclusive divides k evenly. (2) No integers between 2 and k/2 inclusive divides k evenly, and k is greater than 5.
Official Answer and Stats are available only to registered users. Register/ Login.
_________________
Consider me giving KUDOS, if you find my post helpful. If at first you don't succeed, you're running about average. ~Anonymous



Math Expert
Joined: 02 Sep 2009
Posts: 39719

If k is a positive integer, is k a prime number? [#permalink]
Show Tags
19 Jun 2010, 12:35
13
This post received KUDOS
Expert's post
6
This post was BOOKMARKED
If k is a positive integer, is k a prime number?Given: \(k=integer>0\). Question: \(k=prime\)? A prime number is a positive integer with exactly two distinct divisors: 1 and itself. (1) No integer between 2 and \(\sqrt{k}\) inclusive divides k evenly > let's assume \(k\) is not a prime, then there must be some integers \(a\) and \(b\) (\(1<a<k\) and \(1<a<k\)), a factors of \(k\), for which \(ab=k\). As given that \(k\) has no factor between 2 and \(\sqrt{k}\) inclusive, then both factors \(a\) and \(b\) must be more than \(\sqrt{k}\). But it's not possible, as the product of two positive integers more than \(\sqrt{k}\) will yield an integer more than \(k\) (\(ab>k\)). Hence our assumption that \(k\) is not a prime is not true > \(k\) is a prime. Sufficient. (2) No integers between 2 and \(\frac{k}{2}\) inclusive divides k evenly, and k is greater than 5 > the same here : let's assume \(k\) is not a prime, then there must be some integers \(a\) and \(b\) (\(1<a<k\) and \(1<a<k\)), a factors of \(k\), for which \(ab=k\) > \(k=ab\geq{\frac{k^2}{4}}\) (as both \(a\) and \(b\) are more than or equal to \(\frac{k}{2}\), then their product \(ab\), which is \(k\), must be more than or equal to \(\frac{k}{2}*\frac{k}{2}\)) > \(4k\geq{k^2}\) > \(k(4k)\geq{0}\). But this inequality cannot be true as \(4k\) will be negative (as given \(k>5\)) and \(k\) is positive so \(k(4k)\) must be negative not positive or zero. Hence our assumption that \(k\) is not a prime is not true > \(k\) is a prime. Sufficient. Answer: D. P.S. The first statement is basically the way of checking whether some # is a prime: Verifying the primality (checking whether the number is a prime) of a given number \(n\) can be done by trial division, that is to say dividing \(n\) by all integer numbers (primes) smaller than \(\sqrt{n}\), thereby checking whether \(n\) is a multiple of \(m\leq \sqrt{n}\). Example: Verifying the primality of \(161\): \(\sqrt{161}\) is little less than \(13\), from integers from \(2\) to \(13\), \(161\) is divisible by \(7\), hence \(161\) is not prime. Hope it helps.
_________________
New to the Math Forum? Please read this: All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Manager
Joined: 04 Apr 2010
Posts: 162

Re: Is k a prime number? [#permalink]
Show Tags
22 Jun 2010, 11:56
Thanks Bunuel. You have been very helpful +1
_________________
Consider me giving KUDOS, if you find my post helpful. If at first you don't succeed, you're running about average. ~Anonymous



Manager
Joined: 18 Oct 2008
Posts: 191

Re: Is k a prime number? [#permalink]
Show Tags
24 Jun 2010, 09:39
hie Bunuel,
wat does 'divides evenly' means here?



Math Expert
Joined: 02 Sep 2009
Posts: 39719

Re: Is k a prime number? [#permalink]
Show Tags
24 Jun 2010, 09:46



Intern
Joined: 14 Jun 2010
Posts: 16
Location: Singapore
Concentration: Strategy
WE: Information Technology (Consulting)

Re: Is k a prime number? [#permalink]
Show Tags
28 Jun 2010, 01:02
Given: K>0
i. k is odd (coz 2 does not divide k). Second test: Let number be 81 and 83. Try few more numbers if you need... only in case of prime numbers condition (i) is true. Hence, SUFFICIENT.
ii. As done in (i), similarly for (ii), k has to be odd (since it does not divide by 2). Second test: Try 9, 11. The condition (ii) is true only in case of prime numbers! Hence, SUFFICIENT.



Intern
Joined: 23 Aug 2009
Posts: 47

Re: k is a positive number. Is K a prime number? 1) No integers [#permalink]
Show Tags
01 Dec 2011, 17:42
don't we need word "inclusive" in second statement? that's consider k=6  no integer between 2 and 3 divides 6 evenly..



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7446
Location: Pune, India

Re: k is a positive number. Is K a prime number? 1) No integers [#permalink]
Show Tags
02 Dec 2011, 22:17
zura wrote: don't we need word "inclusive" in second statement? that's consider k=6  no integer between 2 and 3 divides 6 evenly.. Yes, I would like the word 'inclusive' if that is what they mean, especially since they have specifically mentioned 'inclusive' in the first statement. It makes me wonder if they are trying to trick me with a 'ha ha, there was no inclusive in the second statement.' As for the actual question, it is very straight forward. The following link discusses the theory of factors which includes that a prime number does not have factors in the range 2 to \(\sqrt{PrimeNumber}\) http://www.veritasprep.com/blog/2010/12 ... tsquares/In case of statement 2, we know that for all n > 5, \(\sqrt{n}\) < n/2 so it basically breaks down and becomes statement 1.
_________________
Karishma Veritas Prep  GMAT Instructor My Blog
Get started with Veritas Prep GMAT On Demand for $199
Veritas Prep Reviews



Senior Manager
Joined: 12 Oct 2011
Posts: 262

Re: k is a positive number. Is K a prime number? 1) No integers [#permalink]
Show Tags
04 Jan 2012, 05:59
Thanks for your wonderful explanations Bunuel and Karishma. You made those daunting problems look so easy.
_________________
Consider KUDOS if you feel the effort's worth it



Manager
Joined: 29 Jul 2011
Posts: 107
Location: United States

Re: k is a positive number. Is K a prime number? 1) No integers [#permalink]
Show Tags
06 Jan 2012, 19:32
Good one, got that one correct! 1. The only way integers between 2 and sqrt(k) will be divisible by k is if K is not prime. Try 2, sqrt(9). Now try 2, sqrt(5) > 5 is not divisible by any integers between 2 and 2.25 (really there is only one integer = 2). Suff. 2. The only way integers between 2 and k/2 will be divisible by k is if K is not prime. try 2 and 6/2. Now try 2, 5/2 and 2, 7/2. Remember integers only. Suff. D.
_________________
I am the master of my fate. I am the captain of my soul. Please consider giving +1 Kudos if deserved!
DS  If negative answer only, still sufficient. No need to find exact solution. PS  Always look at the answers first CR  Read the question stem first, hunt for conclusion SC  Meaning first, Grammar second RC  Mentally connect paragraphs as you proceed. Short = 2min, Long = 34 min



Manager
Joined: 06 Jun 2011
Posts: 147

Re: k is a positive number. Is K a prime number? 1) No integers [#permalink]
Show Tags
07 Jan 2012, 00:20
D. Very helpful analysis by bunuel



Intern
Joined: 11 May 2013
Posts: 8

k is a positive integer. Is k a prime number? [#permalink]
Show Tags
02 Jun 2013, 14:28
From Barron's passkey to the GMAT Third Edition: Page 320 question #5k is a positive integer. Is k a prime number? 1) No integer between 2 and \(\sqrt{k}\) inclusive divides k evenly. 2) No integer between 2 and \(\frac{k}{2}\) inclusive divides k evenly, and k is greater than 5. Statement 1 alone is sufficient since if k is not a prime then k=(m)(n) where m and n must be integers less than k. But this means either m or n must be less than or equal to \(\sqrt{k}\) since if m and n are both larger than \(\sqrt{k}\), (m)(n) is larger than (\(\sqrt{k}\))(\(\sqrt{k}\)) or k. So statement 1 implies k is a prime. Statement 2 alone is also sufficient, since if k=(m)(n) and m and n are both larger than \(\frac{k}{2}\), then (m)(n) is greater than \(\frac{k^2}{2}\); but \(\frac{k^2}{2}\) is greater than k when k is larger than 5. Therefore, if no integer between 2 and \(\frac{k}{2}\) inclusive divides k evenly, then k is a prime.



Manager
Joined: 29 Jun 2011
Posts: 161
WE 1: Information Technology(Retail)

Re: If k is a positive integer, is k a prime number? [#permalink]
Show Tags
03 Sep 2013, 03:22
For sure statment2 requires "Inclusive". E.g) K=8, then between 2 & K/2. There is no interger between 2 & 4 that divides 8.Similary,K=7,then between 2 & K/2 there is no integer that divides K. Hence we require inclusive



Intern
Status: active
Joined: 13 May 2013
Posts: 11

Re: If k is a positive integer, is k a prime number? [#permalink]
Show Tags
03 Sep 2013, 16:28
1
This post received KUDOS
For Number is prime it is not divisible by numbers from 2 to square root of number. k/2 is a bonus, hence D.
_________________
Press Kudos if I was able to help you with questions/solutions/resources
Useful Resources: FlashCards: manhattangmat[dot]com/pdf/FlashCards_Complete_2009.pdf



Intern
Joined: 26 May 2013
Posts: 4

Re: If k is a positive integer, is k a prime number? [#permalink]
Show Tags
13 Nov 2013, 12:54
Hi Bunuel Fir this question if we consider K to be 7, then K/2 is 3.5. The integers between 2 and 3.5 is only 3 and 3 does not divide 3.5 evenly. Secondly if we take K to be 8, K/2 is 4. Integers between 2 and 4 is 3 only, again 4 is divisible by 3 evenly. Hence, second statement does not show anything. I marked A. Please help!!



Math Expert
Joined: 02 Sep 2009
Posts: 39719

Re: If k is a positive integer, is k a prime number? [#permalink]
Show Tags
13 Nov 2013, 12:58



Intern
Joined: 26 May 2013
Posts: 4

Re: If k is a positive integer, is k a prime number? [#permalink]
Show Tags
13 Nov 2013, 15:47
Hi Bunuel
Sorry, I meant not divisible by 3.



Math Expert
Joined: 02 Sep 2009
Posts: 39719

Re: If k is a positive integer, is k a prime number? [#permalink]
Show Tags
13 Nov 2013, 15:56



Intern
Joined: 20 Nov 2012
Posts: 18

Re: If k is a positive integer, is k a prime number? [#permalink]
Show Tags
15 Nov 2013, 06:08
Hi Bunuel,
I have read your explanation regarding statement 2, and then reread it again and again and again.. but I still don't see the answer..
k=ab is pretty clear, but what confuses me is that k=ab>=(k^2)/4.
For example, let's look at a number 100. Then K/2 in this case is 50. If we take two integers a and b which multiply out to give a 100 then a and b, individually, cannot be greater than k/2 or 50 but less...



Math Expert
Joined: 02 Sep 2009
Posts: 39719

Re: If k is a positive integer, is k a prime number? [#permalink]
Show Tags
15 Nov 2013, 08:01
bluecatie1 wrote: Hi Bunuel,
I have read your explanation regarding statement 2, and then reread it again and again and again.. but I still don't see the answer..
k=ab is pretty clear, but what confuses me is that k=ab>=(k^2)/4.
For example, let's look at a number 100. Then K/2 in this case is 50. If we take two integers a and b which multiply out to give a 100 then a and b, individually, cannot be greater than k/2 or 50 but less... Read the highlighted part: (2) No integers between 2 and \(\frac{k}{2}\) inclusive divides k evenly, and k is greater than 5 > the same here : let's assume \(k\) is not a prime, then there must be some integers \(a\) and \(b\) (\(1<a<k\) and \(1<a<k\)), a factors of \(k\), for which \(ab=k\) > \(k=ab\geq{\frac{k^2}{4}}\) ( as both \(a\) and \(b\) are more than or equal to \(\frac{k}{2}\), then their product \(ab\), which is \(k\), must be more than or equal to \(\frac{k}{2}*\frac{k}{2}=\frac{k^2}{4}\)) > \(4k\geq{k^2}\) > \(k(4k)\geq{0}\). But this inequality cannot be true as \(4k\) will be negative (as given \(k>5\)) and \(k\) is positive so \(k(4k)\) must be negative not positive or zero. Hence our assumption that \(k\) is not a prime is not true > \(k\) is a prime. Sufficient.
_________________
New to the Math Forum? Please read this: All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics




Re: If k is a positive integer, is k a prime number?
[#permalink]
15 Nov 2013, 08:01



Go to page
1 2
Next
[ 27 posts ]




