NP Guide (3rd Ed) - Divisibility & Primes Advanced (pg. 117)

[jay.mathew]

I noticed the following question in the Divisibility & Primes Advanced section:

If x is a prime number, what is the value of x?

(1) There are a total of 50 prime numbers between 2 and x, inclusive.
(2) There is no integer n such that x is divisible by n and 1 < n < x.

The answer in the strategy guide is (A). However, when I first attempted to solve the problem, I assumed that x could be negative, so I chose answer choice (E). My question is, when "between 2 and x" is mentioned, can we assume that 2 is the minimum value in the range?

[jay.mathew]

I was also going through the Advanced problem set on pg. 151 of the NP study guide (3rd edition), and stumbled upon the following data sufficiency question which I think raises a similar question:

If x, y, and z are prime numbers and x < y < z, what is the value of x?
(1) xy is even.
(2) xz is even.

Here is how I went about solving the problem:

1.
Statement 1 could have a number of scenarios:
x = -2, y = 2
xy = -4 (even)
x = 2, y = 3
xy = 6 (even)
NOT SUFFICIENT

2.
x = 2, z = 5
xz = 10 (even)
x = -2, z = 3
xz = -6 (even)
NOT SUFFICIENT

Taking the two statements together, we see that x can be either -2, or 2. Since both statements together are still not sufficient to solve the problem, the answer is E.

However, the answer (on pg. 154) is D.

I guess the over-arching issue here is whether we can assume that primes cannot be negative.

[StaceyKoprince]

By definition, prime numbers are positive only. So, if a number is a prime number, it cannot be negative by definition. :)

[jay.mathew]

Thanks again Stacey.

Just to clarify:

Since a prime number is a number that is only divisible by 1 and itself, then this by definition excludes negative numbers, because there can only be 2 factors that make up the prime number?

[esledge]

Hi Jay,

Primes are positive not because of the way the factors are defined, but because "positive" is simply a fundamental quality of a prime number.

I think you are trying to eliminate numbers such as -7 from consideration by thinking of the factors 1, 7, -1, and -7. But if I wanted to be ornery (and I do), I'd point out that +7 is divisible by 1, 7, -1, and -7 too.

So to clarify: All primes must be positive, and their more unique quality is that each prime has exactly two positive divisors: 1 and the prime itself.

[jay.mathew]

Thanks Emily.

[Ben Ku]

I'm glad it worked out!

[bensonhua]

I have a question on the same problem, when it says
If X is a prime number, what is the value of X?

1) there is a total of 50 prime numbers between 2 and x, inclusive

I thought this was insufficient as there are a bunch of prime number between 2 and 50? how could you know which one since it could be any of them?

[Ben Ku]

I have a question on the same problem, when it says
If X is a prime number, what is the value of X?

1) there is a total of 50 prime numbers between 2 and x, inclusive

I thought this was insufficient as there are a bunch of prime number between 2 and 50? how could you know which one since it could be any of them?

Statement (1) is NOT saying "x is a prime number between 2 and 50." (which is I believe how you interpreted it). Instead, it is saying "between 2 and x, there are 50 prime numbers." What that means is that x is a prime number between the 50th and 51st prime number, including the 50th. If P50 represents the 50th prime number, and P51 represents the 51st prime number, then

P50 <= x < P51

The only prime number in that interval is P50, so statement (1) is sufficient.