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

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.

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