Bunuel wrote:

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 \(\frac{-1 + 2}{2} = \frac{1}{2}\).

Answer: B

Hi Bunuel,

This is a great solution.

I was wondering - if at all the solution is possible without knowing the concept around no. of factors for perfect square integers and non-perfect square integers - My try at it says, it is not possible and i wanted to confirm it with an expert.

The way i tried solving it (assuming that one wouldnt know the '# of factors' concept)

1. since a prime no. can be product of two nos. only when they are its factors hence between a and b, one has to be 1 and one has to be the prime no. itself

2. since both are negative and prime nos. are always positive, hence a,b will have to be {-1, -p (i.e. the prime no. p)}

Also as the smallest prime n is = 2, hence between a and b, one will always be more negative than -1, and hence smaller

3. since median for a list of 4 nos. = (2nd no + 3rd no)/2, arranged in ascending order;

for the list {-p, -1, p, some number n} => the median = (-1+p)/2 => p = (2*median)+1

4. Now, that i start back-calculating for p, by using the answer options, the option where p = a prime no. should be the correct answer

5. As both options B and D result in a prime no - i.e. 2 and 5; to be able to choose from one of them - the only way forward is to know the 'the concept around no. of factors for perfect square integers and non-perfect square integers'

Right??

Please confirm. TIA