Author 
Message 
TAGS:

Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 47898

The product of two negative integers, a and b, is a prime nu
[#permalink]
Show Tags
03 Jun 2014, 05:55
Question Stats:
46% (01:23) correct 54% (01:32) wrong based on 564 sessions
HideShow timer Statistics
The product of two negative integers, \(a\) and \(b\), is a prime number \(p\). If \(p\) is the number of factors of \(n\), where \(n\) is NOT a perfect square, what is the value of the median of the four integers \(a\), \(b\), \(p\), and \(n\)?A. 0 B. 1/2 C. 1 D. 3/2 E. 2 Kudos for a correct solution.
Official Answer and Stats are available only to registered users. Register/ Login.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  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




Math Expert
Joined: 02 Sep 2009
Posts: 47898

Re: The product of two negative integers, a and b, is a prime nu
[#permalink]
Show Tags
03 Jun 2014, 05:55
SOLUTIONThe product of two negative integers, \(a\) and \(b\), is a prime number \(p\). If \(p\) is the number of factors of \(n\), where \(n\) is NOT a perfect square, what is the value of the median of the four integers \(a\), \(b\), \(p\), and \(n\)?A. 0 B. 1/2 C. 1 D. 3/2 E. 2 This is a hard questions which tests several number theory concepts. Start from n: we are told that \(n\) is NOT a perfect square. The number of factors of perfect square is odd and all other positive integers have even number of factors. Hence, since \(p\) is the number of factors of \(n\), then \(p\) must be even. We also know that \(p\) is a prime number and since the only even prime is 2, then \(p=2\). Notice here that from this it follows that \(n\) must also be a prime, because only primes have 2 factors: 1 and itself.Next, \(ab=p=2\) implies that \(a=1\) and \(b=2\) or viseversa. So, the set is {2, 1, 2, some prime}, which means that the median is (1 + 2)/2 = 1/2. Answer: B. Theory on Number Properties: mathnumbertheory88376.htmlTips and hints about Number PropertiesDS Number Properties Problems to practice: search.php?search_id=tag&tag_id=38PS Number Properties Problems to practice: search.php?search_id=tag&tag_id=59Please share your number properties tips HERE and get kudos point. Thank you.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  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




Senior Manager
Joined: 13 Jun 2013
Posts: 277

Re: The product of two negative integers, a and b, is a prime nu
[#permalink]
Show Tags
03 Jun 2014, 08:05
Bunuel wrote: The product of two negative integers, \(a\) and \(b\), is a prime number \(p\). If \(p\) is the number of factors of \(n\), where \(n\) is NOT a perfect square, what is the value of the median of the four integers \(a\), \(b\), \(p\), and \(n\)?A. 0 B. 1/2 C. 1 D. 3/2 E. 2 Kudos for a correct solution. all prime numbers are divisible by 1 and itself. now since product of a and b is a prime number P. therefore one of a and b is 1 and other is P. Also, all the perfect square have odd no. of factors. for e.g. a^2 will have 3 factors a,a^2 and 1. Since, n is not a perfect square therefore its no. of factors must be even and the only prime no. which is even is 2. therefore p=2, now the median of numbers 2,1,2 and n will be (1+2)/2= 1/2 hence B



Intern
Joined: 10 Jan 2013
Posts: 39
Concentration: Marketing, Strategy
GMAT 1: 640 Q42 V36 GMAT 2: 680 Q47 V36
WE: Marketing (Transportation)

Re: The product of two negative integers, a and b, is a prime nu
[#permalink]
Show Tags
04 Jun 2014, 10:54
Bunuel wrote: The product of two negative integers, \(a\) and \(b\), is a prime number \(p\). If \(p\) is the number of factors of \(n\), where \(n\) is NOT a perfect square, what is the value of the median of the four integers \(a\), \(b\), \(p\), and \(n\)?A. 0 B. 1/2 C. 1 D. 3/2 E. 2 Kudos for a correct solution. We know that the only way two integers will give us a prime is 1 times a prime number. So A or B is 1 and know both numbers are negative. Since N is not a perfect square it should have 2 factors. Since P=the number of factors, 2, and A or B is 1, then A or B must also be 2. A*B=2 A=1 B=2 P=2 P=N=2 Set of Numbers {2,1,2,2} Median = ((1+2)/2) =1/2 Answer B



Math Expert
Joined: 02 Sep 2009
Posts: 47898

Re: The product of two negative integers, a and b, is a prime nu
[#permalink]
Show Tags
04 Jun 2014, 12:09
GeorgeA023 wrote: Bunuel wrote: The product of two negative integers, \(a\) and \(b\), is a prime number \(p\). If \(p\) is the number of factors of \(n\), where \(n\) is NOT a perfect square, what is the value of the median of the four integers \(a\), \(b\), \(p\), and \(n\)?A. 0 B. 1/2 C. 1 D. 3/2 E. 2 Kudos for a correct solution. We know that the only way two integers will give us a prime is 1 times a prime number. So A or B is 1 and know both numbers are negative. Since N is not a perfect square it should have 2 factors. Since P=the number of factors, 2, and A or B is 1, then A or B must also be 2. A*B=2 A=1 B=2 P=2 P=N=2 Set of Numbers {2,1,2,2} Median = ((1+2)/2) =1/2 Answer BNotice that it's not necessary n to be 2, it could be any other prime as well.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  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



Math Expert
Joined: 02 Sep 2009
Posts: 47898

Re: The product of two negative integers, a and b, is a prime nu
[#permalink]
Show Tags
04 Jun 2014, 12:09
SOLUTIONThe product of two negative integers, \(a\) and \(b\), is a prime number \(p\). If \(p\) is the number of factors of \(n\), where \(n\) is NOT a perfect square, what is the value of the median of the four integers \(a\), \(b\), \(p\), and \(n\)?A. 0 B. 1/2 C. 1 D. 3/2 E. 2 This is a hard questions which tests several number theory concepts. Start from n: we are told that \(n\) is NOT a perfect square. The number of factors of perfect square is odd and all other positive integers have even number of factors. Hence, since \(p\) is the number of factors of \(n\), then \(p\) must be even. We also know that \(p\) is a prime number and since the only even prime is 2, then \(p=2\). Notice here that from this it follows that \(n\) must also be a prime, because only primes have 2 factors: 1 and itself.Next, \(ab=p=2\) implies that \(a=1\) and \(b=2\) or viseversa. So, the set is {2, 1, 2, some prime}, whcih means that the median is (1 + 2)/2 = 1/2. Answer: B. Theory on Number Properties: mathnumbertheory88376.htmlTips and hints about Number PropertiesDS Number Properties Problems to practice: search.php?search_id=tag&tag_id=38PS Number Properties Problems to practice: search.php?search_id=tag&tag_id=59
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  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



Intern
Joined: 10 Jan 2013
Posts: 39
Concentration: Marketing, Strategy
GMAT 1: 640 Q42 V36 GMAT 2: 680 Q47 V36
WE: Marketing (Transportation)

Re: The product of two negative integers, a and b, is a prime nu
[#permalink]
Show Tags
04 Jun 2014, 12:13
Bunuel wrote: GeorgeA023 wrote: Bunuel wrote: The product of two negative integers, \(a\) and \(b\), is a prime number \(p\). If \(p\) is the number of factors of \(n\), where \(n\) is NOT a perfect square, what is the value of the median of the four integers \(a\), \(b\), \(p\), and \(n\)?A. 0 B. 1/2 C. 1 D. 3/2 E. 2 Kudos for a correct solution. We know that the only way two integers will give us a prime is 1 times a prime number. So A or B is 1 and know both numbers are negative. Since N is not a perfect square it should have 2 factors. Since P=the number of factors, 2, and A or B is 1, then A or B must also be 2. A*B=2 A=1 B=2 P=2 P=N=2 Set of Numbers {2,1,2,2} Median = ((1+2)/2) =1/2 Answer BNotice that it's not necessary n to be 2, it could be any other prime as well. Bunuel, Yeah I see that now. When I read it I glanced over the second of. I saw P is the number of factors N and not P is the number of factors of N. So I read it as P=N and not the way it was written.



SVP
Status: The Best Or Nothing
Joined: 27 Dec 2012
Posts: 1835
Location: India
Concentration: General Management, Technology
WE: Information Technology (Computer Software)

Re: The product of two negative integers, a and b, is a prime nu
[#permalink]
Show Tags
05 Jun 2014, 23:20
Assumed numbers: a, b, p, n as 2, 1, 2, 3 2 * 1 = 2 (Prime Number) 3 has 2 factors (1 & 3); It is not a square Median \(= \frac{21}{2} = \frac{1}{2}\) Answer = B
_________________
Kindly press "+1 Kudos" to appreciate



Senior Manager
Joined: 02 Apr 2014
Posts: 484

Re: The product of two negative integers, a and b, is a prime nu
[#permalink]
Show Tags
01 Jan 2018, 07:18
given: 1. a * b = prime number 2. a and b ve numbers
so one of them has to be 1 say a = 1
probable ascending order: { b , a, p , n} since p is number of factors of n, where n is not a perfect square. p cannot be odd (as only perfect square has odd number of factors)
so p is even, so median of {b,a,p,n} = (even  1)/2 = odd / 2 = fraction.
Only option B and D left.
if median were 3/2, then p = 4 => not a prime => so D is out
Answer (B)



Manager
Joined: 16 Jan 2018
Posts: 53

The product of two negative integers, a and b, is a prime nu
[#permalink]
Show Tags
25 Feb 2018, 21:04
Excellent Excellence question!!!
a*b is prime. Since, prime number(P) only has 2 factors 1 and P and any factor of P must contain of product that arrives at P. The only way the product is prime if one of the number is prime (the same number) and other is 1. In this case both negative. so, a = 1 and B = P (or vice versa).
Now, N is not a perfect square so the number of factor has to be even. The only way that p is both even and prime is if it is 2.
so, a =1,B=2 and p =2.
I got kind of stuck here and could not determine n. But was pretty sure it is 2 or 2.
if it is 2 then the set is 2,1,2,2 . median is 1/2.. MY answer if it is 2, then set is 2,2,1,2 median is 3/2. not one of the option..



Intern
Joined: 20 Dec 2017
Posts: 34

The product of two negative integers, a and b, is a prime nu
[#permalink]
Show Tags
26 Feb 2018, 00:12
A really tough question, took me a good 7 min to solve.
The product of two negative integers, a and b, is a prime number p.
If p is a prime number, a is 1 and b is p, or vice versa.
If p is the number of factors of n, where n is NOT a perfect square,
A nonperfect square integer will always have even number of factors. And the only even prime number is 2. Hence a & b is 1 & 2 or vice versa.
what is the value of the median of the four integers a, b, p, and n
We know the following values are present 2, 1, 2, n. Since 2<n and the question is asking about median, we do not need to know the exact value of n. The answer is \((1+2)/2 = 1/2\)
Ans: B




The product of two negative integers, a and b, is a prime nu &nbs
[#permalink]
26 Feb 2018, 00:12






