II wrote:If p is a prime number greater than 2, what is the value of p ?

(1) There are a total of 100 prime numbers between 1 and p+1

(2) There are a total of p prime numbers between 1 and 3912

Thanks.

(1) We could write out a list of the first 100 primes. Since there are 100 primes between 1 and p+1, p is the 100th prime on our list: sufficient.

(2) We could write out all the primes between 1 and 3912 and count them: sufficient.

Each of (1) and (2) is sufficient on its own: choose (D).

Easy as that !
Thanks Stuart ...

stuart isnt the answer b,
in the sense says there are 3 prime numbers between 2 and 11 (p+1) ,but there are also 3 prime numbers between 2 and 8 ,2 and 9 ,2 and 10 , 2 and 11(since 8,9 and 10 are not prime)..so p could be 7 ,8,9 or 10 since question specifically asks for value of p.

gmatguy16 wrote:stuart isnt the answer b,
in the sense says there are 3 prime numbers between 2 and 11 (p+1) ,but there are also 3 prime numbers between 2 and 8 ,2 and 9 ,2 and 10 , 2 and 11(since 8,9 and 10 are not prime)..so p could be 7 ,8,9 or 10 since question specifically asks for value of p.

We know that p is a prime number, so we also know that p+1 isn't a prime number (the only consecutive primes are 2 and 3, and p certainly isn't 2 based on other info we have).

So, in your example, the only number that would be a possible match would be p=7 and p+1=8.

If we hadn't known that p was prime, then you'd be right, there would potentially be multiple values (we'd have to actually count them out to tell).

Stuart, is it not necessary to have "unique" value of p from both the statements?

One can write first 100 primes and can find P from statement 1. Meanwhile one can write all the primes between 1 and 3912. My question is what is the surety that the answers from both the statements give a unique p? How one can cross check for such big problems or is not necessary that two statements should point to the unique single number? I am somewhat confused....

microke wrote:Stuart, is it not necessary to have "unique" value of p from both the statements?

One can write first 100 primes and can find P from statement 1. Meanwhile one can write all the primes between 1 and 3912. My question is what is the surety that the answers from both the statements give a unique p? How one can cross check for such big problems or is not necessary that two statements should point to the unique single number? I am somewhat confused....

If a statement is sufficient by itself, you NEVER have to worry about the statements together.

By design, any time that each statement gives you enough information, the two statements always agree on the answer. Otherwise, when you combined the statements (it always has to be possible to combine them, even though it's not always necessary), you'd get a null answer to the question - and there's always at least one answer.

Here are a couple of examples of what you would NEVER see on the GMAT:

What's the value of x?

1) x + 14 = 23

2) x - 14 = 23

Each statement gives you a unique value for x, but they give conflicting values. Therefore, the above is an impossible (i.e. it would NEVER appear) GMAT question.

Is x > 0?

1) x > 20

2) x < -3

Looked at separately, each statement gives you a definite answer: from (1) we get a definite "yes" and from (2) we get a definite "no". However, on the GMAT that will NEVER happen, since taken together there are no possible values for x (i.e. there's no number that's BOTH greater than 20 AND less than -3). Hence, this would be another impossible (i.e. it would NEVER appear) GMAT question.

On a side note, you can sometimes use the above rule to double check your work. If you decided that each statement was sufficient but you notice that each one gave you a different answer, then you can be certain that you made a mistake along the way.

Thanks Stuart, for clarifying the important concept in DS. Now I understood GMAC philosophy on DS. Very valuble input.....

The part I stumbled on in this question is that nowhere does it say that the prime numbers need to be unique. I guess it can be somewhat inferred, but I was thinking what if all the prime numbers are all the same, who knows what p can ever be?

ohwell wrote:The part I stumbled on in this question is that nowhere does it say that the prime numbers need to be unique. I guess it can be somewhat inferred, but I was thinking what if all the prime numbers are all the same, who knows what p can ever be?

Whenever a question talks about "the numbers between x and y", it's referring to the number line, on which each number appears exactly once.

Stuart Kovinsky wrote:

II wrote:If p is a prime number greater than 2, what is the value of p ?

(1) There are a total of 100 prime numbers between 1 and p+1

(2) There are a total of p prime numbers between 1 and 3912

Thanks.

(1) We could write out a list of the first 100 primes. Since there are 100 primes between 1 and p+1, p is the 100th prime on our list: sufficient.

(2) We could write out all the primes between 1 and 3912 and count them: sufficient.

Each of (1) and (2) is sufficient on its own: choose (D).

Good explanation

I just hope I'm never asked to find the sum of the first 100 prime numbers!

As Stuart rightly put it, we need not find the exact answer in DS (unlike PS). All we need to check is whether the statement(s) give us a 100% unique (consistent) answer.

II wrote:If p is a prime number greater than 2, what is the value of p ?

(1) There are a total of 100 prime numbers between 1 and p+1

(2) There are a total of p prime numbers between 1 and 3912

We are given that p is a prime number greater than 2 and we need to determine the value of p.

Note that even though we are asked for the value of p, we actually need to determine whether the value of p is unique. If we can determine from the given statements that p is unique, then the statement(s) will be sufficient. We do not have to actually determine the value of p, even though it would be possible.

Statement One Alone:

There are a total of 100 prime numbers between 1 and p + 1.

If there are exactly 100 prime numbers between 1 and p + 1, then there are exactly 100 prime numbers in the list: 2, 3, 5, 7, 11, 13, ..., p. Whatever value p is, p must be unique. It is the 100th number in the list. Statement one alone is sufficient. We can eliminate answer choices B, C, and E.

Statement Two Alone:

There are a total of p prime numbers between 1 and 3,912.

It is a fact that between two distinct positive integers, there must be a unique number of primes. For example, between 1 and 10 inclusive there are exactly 4 primes: 2, 3, 5, 7. There can't be 3 primes or 5 primes between 1 and 10. Therefore, if there are exactly p prime numbers between 1 and 3,912, p must be unique, even if we don't know its exact value. Statement two alone is also sufficient.

Answer: D