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

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Math Expert
Joined: 02 Sep 2009
Posts: 43292

Kudos [?]: 139150 [7], given: 12777

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

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03 Jun 2014, 04:55
7
KUDOS
Expert's post
12
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BOOKMARKED
00:00

Difficulty:

95% (hard)

Question Stats:

46% (01:19) correct 54% (01:31) wrong based on 430 sessions

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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

_________________

Kudos [?]: 139150 [7], given: 12777

Math Expert
Joined: 02 Sep 2009
Posts: 43292

Kudos [?]: 139150 [1], given: 12777

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

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03 Jun 2014, 04:55
1
KUDOS
Expert's post
6
This post was
BOOKMARKED
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.

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.
_________________

Kudos [?]: 139150 [1], given: 12777

Senior Manager
Joined: 13 Jun 2013
Posts: 278

Kudos [?]: 499 [2], given: 13

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

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03 Jun 2014, 07:05
2
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.

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

Kudos [?]: 499 [2], given: 13

Intern
Joined: 10 Jan 2013
Posts: 39

Kudos [?]: 44 [1], given: 13

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]

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04 Jun 2014, 09:54
1
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

Kudos [?]: 44 [1], given: 13

Math Expert
Joined: 02 Sep 2009
Posts: 43292

Kudos [?]: 139150 [0], given: 12777

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

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04 Jun 2014, 11:09
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

Notice that it's not necessary n to be 2, it could be any other prime as well.
_________________

Kudos [?]: 139150 [0], given: 12777

Math Expert
Joined: 02 Sep 2009
Posts: 43292

Kudos [?]: 139150 [1], given: 12777

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

### Show Tags

04 Jun 2014, 11:09
1
KUDOS
Expert's post
1
This post was
BOOKMARKED
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.

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
_________________

Kudos [?]: 139150 [1], given: 12777

Intern
Joined: 10 Jan 2013
Posts: 39

Kudos [?]: 44 [0], given: 13

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]

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04 Jun 2014, 11:13
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

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.

Kudos [?]: 44 [0], given: 13

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Status: The Best Or Nothing
Joined: 27 Dec 2012
Posts: 1844

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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]

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05 Jun 2014, 22:20
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 $$= \frac{2-1}{2} = \frac{1}{2}$$

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Kudos [?]: 2859 [0], given: 193

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Joined: 02 Apr 2014
Posts: 343

Kudos [?]: 30 [0], given: 1091

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

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01 Jan 2018, 06:18
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

Kudos [?]: 30 [0], given: 1091

Re: The product of two negative integers, a and b, is a prime nu   [#permalink] 01 Jan 2018, 06:18
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