M30-12

Question

The product of two negative integers,  a  and  b , is a prime number  p . If  p  is the number of positive factors of a positive integer  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

Hoehenheim · Answer

Hello there Bunuel! This looks like a goldmine for DS questions. Could you point me as to where this particular theory was extracted from? Is there a link for such concise summaries of various concepts?

Bunuel · Answer

2. Properties of Integers 
     Theory   
 Math: Number Theory 
 Number Properties: Tips and hints 
 Advanced Number Properties on GMAT (All 5 parts) 
 Do you make these 3 mistakes in GMAT Even-Odd Questions? 
 Everything about Factorials on the GMAT 
 Solving Complex Problems Using Number Line 
     Questions   
 Tough New Set: Number Properties Set 
 Power of a Number in a Factorial Problems 
 Number Properties Ability Quiz 
 Special Numbers and Sequences Problems 
 The E-GMAT Number Properties Knockout: 10 Great 700+ Ques for you! 
 Trailing Zeros Problems 
 DS Questions 
 PS Questions

For other subjects:
 ALL YOU NEED FOR QUANT ! ! ! 
 Ultimate GMAT Quantitative Megathread

Hope this helps.

mellowmev · Answer

I think this is a high-quality question and I agree with explanation.

LessgotoM7 · Answer

Hey  Bunuel ,
Lets say a = -1, b = -5, hence p = 5. Now n can be some prime number which has 5 factors.
Median of the set {-5, -1, 5, n} would be 2. Am I missing something?

Thanks

Bunuel · Answer

A prime number has only two positive factors: 1 and itself! If a number has 5 positive factors, it must be a perfect square. For example, 2^4 has five positive factors and is the square of an integer. Please review the solution more carefully.

LessgotoM7 · Answer

Thanks,
Was typo at my end that n is a prime number. But yes, number with 5 positive factors will be a perfect square.

ms27 · Answer

I like the solution - it’s helpful.

Vikeesh · Answer

I like the solution - it’s helpful.

Sowmya_10 · Answer

Suppose I take the set to be {-3,-1, 3, some prime} then my median would be 1. In that case answer C would be the correct answer choice

Bunuel · Answer

This set is not valid because a prime number cannot have 3 factors. By definition, a prime has exactly 2 positive factors: 1 and itself.

SaiJ1011 · Answer

This is a great question that’s helpful for learning. Great conceptual question

abhudayaharp · Answer

This is a great question that’s helpful for learning and I like the solution - it’s helpful.

stinson07 · Answer

Hi,

Yes, when you arrange the numbers for finding median, the correct order will be {-2, -1, 2, some prime number}.

shivampaul · Answer

I like the solution - it’s helpful.

nandini14 · Answer

If I take a=1, b=2, hence p=2 and I can take n=3 then my median will be 2; can you please tell me what is incorrect in my approach?

Bunuel · Answer

Please read the question carefully:

The product of two   negative   integers,  a  and  b , is a prime number  p . If  p  is the number of positive factors of a positive integer  n , where  n  is NOT a perfect square, what is the value of the median of the four integers  a ,  b ,  p , and  n ?

3D · Answer

This is a great question that’s helpful for learning.

nandini14 · Answer

But it's nowhere mentioned in the question that n is also a prime number? In this set n can be 6 as well and its not a perfect square

Bunuel · Answer

Please, read the soliton carefully:

Let's begin with the positive integer  n , which we know is not a perfect square. We can recall that positive perfect squares have an odd number of factors while all other positive integers have an even number of factors. Hence, the number of factors of  n , which is  p , must be even. We are also told that  p  is a prime number, and since the only even prime number is  2 , it follows that  p=2 . Moreover,  since only prime numbers have exactly  2  factors (1 and itself),  n  must also be a prime number.

M30-12

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