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

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. 1/2
C. 1
D. 3/2
E. 2

Bunuel · Accepted Answer

Official Solution:

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. \(2\)

This question covers several concepts in number theory and is quite challenging.

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.

Given that \(ab=p=2\), we have either \(a=-1\) and \(b=-2\), or vice versa. Therefore, the set of four integers is {-2, -1, 2, some prime}. The median of this set is \(\frac{-1+2}{2}=\frac{1}{2}\).

Answer: B

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manpreetsingh86 · Answer

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

GeorgeA023 · Answer

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

Answer B

Bunuel · Answer

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

Bunuel · Answer

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.

Answer: B.

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GeorgeA023 · Answer

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.

PareshGmat · Answer

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}

Answer = B

hellosanthosh2k2 · Answer

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

Answer (B)

krish76 · Answer

Excellent Excellence question!!!

a*b is prime. 
Since, prime number(P) only has 2 factors 1 and P and any factor of P must contain of product that arrives at P.
The only way the product is prime if one of the number is prime (the same number) and other is 1. In this case both negative.
so, a = -1 and B = -P (or vice versa).

Now, N is not a perfect square so the number of factor has to be even. The only way that p is both even and prime is if it is 2.

so, a =-1,B=-2 and p =2.

I got kind of stuck here and could not determine n. But was pretty sure it is -2 or 2.

if it is 2 then the set is -2,-1,2,2 . median is 1/2.. MY answer
if it is -2, then set is -2,-2,-1,2 median is -3/2. not one of the option..

Wongderful · Answer

A really tough question, took me a good 7 min to solve.

The product of two negative integers, a and b, is a prime number p.

If p is a prime number, a is -1 and b is -p, or vice versa.

If p is the number of factors of n, where n is NOT a perfect square,

A non-perfect square integer will always have even number of factors. And the only even prime number is 2. Hence a & b is -1 & -2 or vice versa.

what is the value of the median of the four integers a, b, p, and n

We know the following values are present -2, -1, 2, n.
Since 2<n and the question is asking about median, we do not need to know the exact value of n.
The answer is  (-1+2)/2 = 1/2

Ans: B

Sneha2021 · Answer

How can we deduce 2<n?
Why can't n=1 ?

Mayank452 · Answer

The question says that n is not a perfect square. 1 is a perfect square.

bumpbot · Answer

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

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