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

Stem and the statements are ALWAYS providing us with correct information.

If it turns out that the quantity of primes between 1 and 3912 is not a prime number itself, this will mean that the question is flawed. GMAT wouldn't give us such question then.

Stem says p is a prime number.
Statement (2) says that "there are a total of p prime numbers between 1 and 3912". So yes the # of primes between 1 and 3912 MUST be prime number itself. We don't know what number it is, but we can calculate it, hence we can calculate p, hence (2) is also sufficient.

Hope it's clear.
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1: p has to be 100th prime number starting from 2 (1st prime number) sufficient.
2: all the prime numbers between 1 and 3912 can be found and the number of prime number will be what we need. Sufficient.
Hence D.

PS: This is a DS question so we do not need to calculate the p till end. calculation will take some time but no need to do.

Using Stmt 1, it's pretty straight forward.

Using Stmt 2, let n = no. of prime numbers between 1 and 3912. How can we be sure that n is a prime number and it is equal to p? Do we have to assume so in questions like these?

Why can't p+1 be the 100th prime number ?

As p is prime more than 2 (given), then p+1 is even so it can not be prime.
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statement A is not sufficient...consider this: how many prime numbers are between 1 and 14 and between 1 and 16,,,same number..you can not conclude anything from statement like there 100 prime less than p+1...
B is sufficient..

OA for this question is D, not B.

Statement (1) is says that there are 100 primes in the range between 1 and P+1, so this statement basically says that P is the 100th prime --> we can determine the single numerical value of P (we can find 100th prime). As this is DS question no matter what the actual value of P is, the fact that we can find it, is already sufficient.
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Statement 1:
Listing the first 100 prime numbers and getting the 100th prime will give us p

Statement 2:
Listing all the prime numbers from 1 to 3912 and counting them will give us p

Answer: D

Incidentally, Bunuel I ran the wolfram to find the number of primes between 1 & 3912. It said 541.

Not to see if 541 is prime or not. We have the closest square root of 541 less than 25 ( considering 625). Thus, we need to check for
2- No
3-No
5-- No
7- No
9-- No ( sum of digits = 10)
11- No
13- No
17- No
19- No
23- No Hence 541 is prime as predicted by you. ( Ohh. by the way, I did countercheck with wolfram again & its correct)

It's good to see the principles in action.

Thanks,
Ankush.

(2) must be sufficient, as there is obviously some fixed number of primes between 1 and 3912. we don't care what that number is, because it's clear that there's only one such number (the number of primes in a fixed range isn't about to change anytime soon).

(1) also sufficient: p is a prime number, so:
if p is the 100th prime, then there are 100 primes - viz., the first 100 primes - between 1 and p + 1.
if p is the 101th prime or later, then there are 101 or more primes, so that's no good.
if p is the 99th prime or earlier, then there are 99 or fewer primes; also no good.
therefore, p is the 100th prime.

answer = d

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
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Bunuel chetan2u niks18

Is my below understanding correct for highlighted text?



Eg 1: 2 is a prime number, numbers that are prime between 1 and 2 is 0

Eg 2: 3 is a prime number, numbers that are prime between 1 and 3 is 1 (2)

Eg 3: 5 is a prime number, numbers that are prime between 1 and 5 is 2 (1,3)

Does the question ask the value in bracket is UNIQUE ? (2 or 1,3) I did not understand significance of P+1 in St 1
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Hi adkikani

First of all note that 1 is not prime.

let's assume p to be 5, so p+1=6

now I can say that there are 3 prime numbers between 1 & 6 (2,3 & 5), which is same as saying there are 3 prime numbers between 1 & p+1, where p is a prime number.

So in this question, we have 100 prime numbers between 1 & p+1, so you can actually count 100 prime numbers starting with 2. the 100th prime number will be p

Answer : D.

Statement 1 : Is obvious , you can actually calculate manually and find out the value of P ( though you don't need to coz we know that there will be a unique value ).
Statement 2 : Again , we understand that there will be a unique value for P between 1 and 3912 which we don't need to sit down and calculate coz it is going to be a unique value hence arriving at option D as the answer.

Pls comment in case you disagree with the logic or have a better/shorter way to solve this question , else Kudos would be nice !

Let's take prime numbers between
1 to 14 and 1 to 15
The value of p changes.
The main aim is to find whether we are able to find p value (or)
Whether to find the p value is always same!
Bunuel

Posted from my mobile device

14 and 15 are not prime numbers.

The upper limit of the series of numbers is "p+1", i.e. the upper limit is the number p (a prime number) plus one.

E.g. If you have p=6 prime numbers, then the last number of the series is equal to 6+1=7.

Where are you getting 14 and 15? Perhaps misunderstood your question.

Forget the conventional way to solve DS questions.

We will solve this DS question using the variable approach.

DS question with 1 variable: Let the original condition in a DS question contain 1 variable. Now, 1 variable would generally require 1 equation to give us the value of the variables.

We know that each condition would usually give us an equation, and since we need 1 equation to match the number of variables and equations in the original condition, the equal number of equations and variables should logically lead to answer D.

To master the Variable Approach, visit https://www.mathrevolution.com and check our lessons and proven techniques to score high in DS questions.

Let’s apply the 3 steps suggested previously. [Watch lessons on our website to master these 3 steps]

Step 1 of the Variable Approach: Modifying and rechecking the original condition and the question.

We have to find the value of 'p'

=> Given that 'p' is a prime number greater than '2'.

Second and the third step of Variable Approach: From the original condition, we have 1 variable (a). To match the number of variables with the number of equations, we need 1 equation. Since conditions (1) and (2) will provide 1 equation each, D would most likely be the answer.

Let’s take a look at each condition.

Condition(1) tells us that there are a total of 100 prime numbers between 1 and p+1.

=> This means 'p' will be the 100^th prime number.

Since the answer is unique, the condition is sufficient by CMT 2.


Condition(2) tells us that there are a total of p prime numbers between 1 and 3,912

=> Listing all the prime numbers between 1 and 3,912 and then counting them will give us a value of 'p'

Since the answer is unique, the condition is sufficient by CMT 2.

Both conditions (1) and (2) alone are sufficient.

So, D is the correct answer.

Answer: D

What do I understand the question is: P is the number of prime numbers greater than 2? What is the value of P?

(1) It Says to total numbers of prime numbers within P+1. Sufficient.

(2) It is possible to count total number of prime numbers. Sufficient.

Ans. D

Fun math fact: The 100th prime is 541, so p = 541.

This DS question is a great example of why you can't always try to calculate the answer—unless you want to waste a ton of time and risk not finishing the section.

You only have to know that there can only be one answer for both conditions 1 and 2 individually (Choice D), since that is the definition of sufficiency. On DS, it's important to remember that you don't have to know precisely what the answer is, only whether there is more than one answer, and it would take most mere mortals—me included—too long to calculate the value of p under time pressure. Even a (theoretical) calculator wouldn't be much help for this one!