For how many prime numbers p, is p^2 + 3p - 1 a prime number?

Question

A. 0
B. 1
C. 2
D. 3
E. 4

Are You Up For the Challenge: 700 Level Questions

nick1816 · Accepted Answer

Any prime number greater than 3 is in the form of 6k±1

1. when p=6k+1, where k is an integer

p^2 + 3p - 1= 36k^2+12k+1+18k+3-1= 36k^2+30k+3= 3(12k^2+10k+1)  
hence, it will always a multiple of 3

2. when p=6k-1, where k is an integer

p^2 + 3p - 1= 36k^2-12k+1+18k-3-1= 36k^2+6k-3= 3(12k^2+2k-1)  
hence, it will always a multiple of 3

p^2 + 3p - 1  can never be a prime number, if p is a prime number greater than 3

Now we can check for p=2 and p=3

when p=2,  p^2 + 3p - 1 = 9 (not a prime number)

when p=3,  p^2 + 3p -  1= 17 (prime number)

B

GMATinsight · Answer

@p = 2, p^2 + 3p - 1=2+6-1 = 9 NOT Prime
 @p = 3, p^2 + 3p - 1=9+9-1 = 17 Prime 
@p = 5, p^2 + 3p - 1=25+15-1 = 39 NOT Prime
@p = 7, p^2 + 3p - 1=49+21-1 = 69 NOT Prime
@p = 11, p^2 + 3p - 1=121+33-1 = 153 NOT Prime

For any prime value of p other than 3, the value of expression p^2 + 3p - 1 is always divisible by 3 hence there will NOT be any next value of p that satisfies the given constraint

Answer: Option B

GMATWhizTeam · Answer

Solution

• The expression  p^2 + 3p – 1  is given as summation of 3 terms, we will try to re-express it as product so that identifying the prime nature will be easier.
• On looking at the given expression we can observe that we can easily manipulate it in an expression containing  (p+1)^2 
• Adding as well as subtracting 1 from the expression, we get:
 o  p^2 + 3p – 1 = p^2 + 2p +1 + p -1-1 = (p+1)^2 + p-2   
• On further adding and subtracting 1 in the above expression, we get,
 o  (p+1)^2 + p-2 = (p+1)^2 + p+1 -2-1 = (p+1)^2 +(p+1)-3 = (p+1)(p+2) -3 
   p, p+1  and  p+2  are consecutive integers.  
• We know that 1 out of every three consecutive integers must be a multiple of 3.
 o If  p+1  or  p+2  are multiple of 3 then we can take 3 common out of  (p+1)(p+2) -3  and it won’t be a prime.
o Thus,  (p+1)(p+2) -3  will be prime only if p is a multiple of 3.
  As p is a prime only such possibility is  p= 3 .    
Thus, the correct answer is  Option B .

Alternate Solution,  
 • As p is a prime it can be either 2 ,3 or it can be represented as 6n+1 or 6n -1, where n is positive integer. So, there will be following four cases:
 o Case 1: If   p = 6n + 1 
   p^2 + 3p – 1 = (6n+1)^2 + 3*(6n+1) – 1 = 36n^2+ 12n + 1 + 3*(6n+1) -1 = 3*(12n^2 + 10n + 1) , it is divisible by 3 hence, this can’t be a prime. 
o Case 2: If  p = 6n – 1 
   p^2 + 3p – 1 = (6n-1)^2 + 3*(6n-1) – 1 = 36n^2- 12n + 1 + 3*(6n-1) -1 = 3*(12n^2 + 2n - 1) , it is also divisible by 3 hence, this can’t be a prime. 
o Case 3: If  p = 2 
   p^2 + 3p – 1 = (2)^2 + 3*(2) – 1 = 9 , which is not a prime. 
o Case 4: If  p = 3 
   p^2 + 3p – 1 = (3)^2 + 3*(3) – 1 = 17 , which is a prime.   
Hence, only for  p = 3  the given expression is a prime.
Thus, the correct answer is  Option B .

bumpbot · Answer

Automated notice from GMAT Club BumpBot:

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

This post was generated automatically.

For how many prime numbers p, is p^2 + 3p - 1 a prime number?

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

For how many prime numbers p, is p^2 + 3p - 1 a prime number?

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards