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The product of two negative integers, a and b, is a prime nu

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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.

[Reveal] Spoiler: OA

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Re: The product of two negative integers, a and b, is a prime nu [#permalink]

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New post 03 Jun 2014, 05:55
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SOLUTION

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

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 vise-versa.

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: math-number-theory-88376.html
Tips and hints about Number Properties

DS Number Properties Problems to practice: search.php?search_id=tag&tag_id=38
PS Number Properties Problems to practice: search.php?search_id=tag&tag_id=59

Please share your number properties tips HERE and get kudos point. Thank you.
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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

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Re: The product of two negative integers, a and b, is a prime nu [#permalink]

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New post 03 Jun 2014, 08:05
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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

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Re: The product of two negative integers, a and b, is a prime nu [#permalink]

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New post 04 Jun 2014, 10:54
1
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KUDOS
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

Kudos [?]: 43 [1], given: 13

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Re: The product of two negative integers, a and b, is a prime nu [#permalink]

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New post 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 B


Notice that it's not necessary n to be 2, it could be any other prime as well.
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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?
Extra-hard Quant Tests with Brilliant Analytics

Kudos [?]: 129127 [0], given: 12194

Expert Post
1 KUDOS received
Math Expert
User avatar
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Joined: 02 Sep 2009
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Kudos [?]: 129127 [1], given: 12194

Re: The product of two negative integers, a and b, is a prime nu [#permalink]

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New post 04 Jun 2014, 12:09
1
This post received
KUDOS
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SOLUTION

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

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 vise-versa.

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: math-number-theory-88376.html
Tips and hints about Number Properties

DS Number Properties Problems to practice: search.php?search_id=tag&tag_id=38
PS 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?
Extra-hard Quant Tests with Brilliant Analytics

Kudos [?]: 129127 [1], given: 12194

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Joined: 10 Jan 2013
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Kudos [?]: 43 [0], given: 13

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GMAT 1: 640 Q42 V36
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Re: The product of two negative integers, a and b, is a prime nu [#permalink]

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New post 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 B


Notice 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.

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Re: The product of two negative integers, a and b, is a prime nu [#permalink]

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New post 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 \(= \frac{2-1}{2} = \frac{1}{2}\)

Answer = B
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Re: The product of two negative integers, a and b, is a prime nu [#permalink]

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Re: The product of two negative integers, a and b, is a prime nu [#permalink]

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Re: The product of two negative integers, a and b, is a prime nu   [#permalink] 10 Sep 2017, 08:34
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