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# M30-12

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Math Expert
Joined: 02 Sep 2009
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16 Sep 2014, 00:45
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Difficulty:

85% (hard)

Question Stats:

50% (01:58) correct 50% (01:31) wrong based on 112 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. $$\frac{1}{2}$$
C. $$1$$
D. $$\frac{3}{2}$$
E. $$2$$

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16 Sep 2014, 00:45
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Official 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. $$\frac{1}{2}$$
C. $$1$$
D. $$\frac{3}{2}$$
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 $$\frac{-1 + 2}{2} = \frac{1}{2}$$.

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10 Jan 2015, 23:55
I think this question is good and helpful.
Nice question covering 2-3 concepts
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08 Apr 2015, 06:34
if the set is {-1,-2,2,some prime} then the median would be 0. Am I missing out something?
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08 Apr 2015, 06:39
sachin2040 wrote:
if the set is {-1,-2,2,some prime} then the median would be 0. Am I missing out something?

The median of a set with even number of elements is the average of two middle terms when arranged in ascending /descending order. The set in ascending order is {-2, -1, 2, some prime} --> median = (-1 + 2)/2 = 1/2.
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08 Apr 2015, 06:44
Thanks Bunuel, that was silly on my part.....
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28 Apr 2016, 06:37
great concept question!
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28 May 2016, 22:04
Notice here that from this it follows that n must also be a prime, because only primes have 2 factors: 1 and itself.

Can n be negative here? Say n = -1, its factors would be 1 and -1.
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30 May 2016, 13:18
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sushruthav wrote:
Notice here that from this it follows that n must also be a prime, because only primes have 2 factors: 1 and itself.

Can n be negative here? Say n = -1, its factors would be 1 and -1.

No, a factor is a positive divisor.
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26 Jun 2016, 05:51
A very good question ! +1 Kudos
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01 Aug 2016, 13:09
I think this is a high-quality question and I agree with explanation.
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04 Aug 2016, 16:25
Bunuel wrote:
sushruthav wrote:
Notice here that from this it follows that n must also be a prime, because only primes have 2 factors: 1 and itself.

Can n be negative here? Say n = -1, its factors would be 1 and -1.

No, a factor is a positive divisor.

Then, Suppose, 3 is the number of factors of N.
Can we say, N is a perfect Square?
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05 Aug 2016, 03:30
nahid78 wrote:
Bunuel wrote:
sushruthav wrote:
Notice here that from this it follows that n must also be a prime, because only primes have 2 factors: 1 and itself.

Can n be negative here? Say n = -1, its factors would be 1 and -1.

No, a factor is a positive divisor.

Then, Suppose, 3 is the number of factors of N.
Can we say, N is a perfect Square?

Yes.

1. The number of distinct factors of a perfect square is ALWAYS ODD. The reverse is also true: if a number has the odd number of distinct factors then it's a perfect square;

2. The sum of distinct factors of a perfect square is ALWAYS ODD. The reverse is NOT always true: a number may have the odd sum of its distinct factors and not be a perfect square. For example: 2, 8, 18 or 50;

3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. The reverse is also true: if a number has an ODD number of Odd-factors, and EVEN number of Even-factors then it's a perfect square. For example: odd factors of 36 are 1, 3 and 9 (3 odd factor) and even factors are 2, 4, 6, 12, 18 and 36 (6 even factors);

4. Perfect square always has even powers of its prime factors. The reverse is also true: if a number has even powers of its prime factors then it's a perfect square. For example: $$36=2^2*3^2$$, powers of prime factors 2 and 3 are even.

Hope it helps.
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06 Oct 2016, 01:29
Bunuel wrote:
So, the set is {-2, -1, 2, some prime}, which means that the median is $$\frac{-1 + 2}{2} = \frac{1}{2}$$. Answer: B

Didn't Get It!
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06 Oct 2016, 03:53
RichaChampion wrote:
Bunuel wrote:
So, the set is {-2, -1, 2, some prime}, which means that the median is $$\frac{-1 + 2}{2} = \frac{1}{2}$$. Answer: B

Didn't Get It!

The median is the average of two middle terms, when arranged in ascending/descending order. So, the median of {-2, -1, 2, some prime} is $$\frac{-1 + 2}{2} = \frac{1}{2}$$ (regardless of the unknown prime there).
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17 Nov 2016, 14:41
Bunuel wrote:
sushruthav wrote:
Notice here that from this it follows that n must also be a prime, because only primes have 2 factors: 1 and itself.

Can n be negative here? Say n = -1, its factors would be 1 and -1.

No, a factor is a positive divisor.

Could you please explain a bit more regarding factors of negative number? Can n be -2 in this case having two factors 1 and 2? From my understanding if number of factors of a number are given the number could be either positive or negative.
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19 Nov 2016, 00:12
Great Question. Very helpful. +1 Kudos.
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09 Jan 2017, 19:25
I think this is a high-quality question and I agree with explanation.
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21 Apr 2018, 03:08
Since a and b can be -1 and -2 or vice versa, are 1/2 and 0 not a possible answer? How did we assume that a=-2 and b=-1?
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21 Apr 2018, 04:16
prkohli wrote:
Since a and b can be -1 and -2 or vice versa, are 1/2 and 0 not a possible answer? How did we assume that a=-2 and b=-1?

Your question is not clear. How are you getting 0 or 1/2 for the median? What set do you have giving those values of median ?

We got that $$a=-1$$ and $$b=-2$$ or vise-versa.

So, the set is {-2, -1, p = 2, n = some prime}, which means that the median is (-1 + 2)/2 = 1/2.
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# M30-12

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