Is the prime number q equal to 23? (1) r = 2p + 1, where p is prime

Question

Is the prime number q equal to 23?

(1) r = 2p + 1, where p is prime
(2) r = 3q + 2, where q is prime

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shameekv1989 · Answer

Is the prime number q equal to 23?

(1) r = 2p + 1, where p is prime
(2) r = 3q + 2, where q is prime

I) No info about q -> Insufficient
II) r = 3q+2 => For q to be 23 - r has to be 71. Though r can be 53 for which q will be 17 - another prime number - Insufficient

Combining, 
r = 2p+1 and r = 3q+2

For q to be 23 r has to be 71. But if r is 71 p has to be 35 which is not prime number - a prerequisite for "p"

Hence q can not be 23. Sufficient.

Answer C

tam87 · Answer

S1: not suff as no info about q
S2: 
r = 3q + 2, where q is prime

q can be any prime number and not necessarily 23 - hence, not sufficient

Combine:

r = 2p + 1 = 3q + 2 where both p and q are prime
now, if q = 23, then 2p + 1 = 71 implying p = 35
but p cannot be 35 as p is a prime and 35 is composite

therefore, q cannot be equal to 23 (option C)

GMATWhizTeam · Answer

Solution 
 Step 1: Analyse Question Stem 
 • q is a prime number.
• We need to find if  q = 23  
 Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE 
 Statement 1:  r = 2p + 1, where p is prime
 • In this statement, nothing is mentioned regarding q. So we cannot conclude anything about the value of q. 
Hence, statement 1 is NOT sufficient and we can eliminate answer Options A and D.
 Statement 2:  r = 3q + 2, where q is prime
 • According to this statement:  3q = r – 2 
• However, we don’t know r so we cannot conclude about q. For example,
 o If  r = 71  then  ⟹ 3q = 71- 2 = 69⟹ q = 23 
o If  r = 11  then  ⟹ 3q = 11-2 = 9⟹q =  3  
Hence, statement 2 is also NOT sufficient and we can eliminate answer Option B.
 Step 3: Analyse Statements by combining. 
 • From statement 1:  r = 2p + 1 , where p is prime
• From statement 2:  r = 3q + 2 , where q is prime
• On combining both statements, we get,
 o  2p + 1 = 3q + 2 ⟹ 3q = 2p – 1 
o Now,  if   q = 23   then  3q =2p – 1 = 69 ⟹ 2p = 70 ⟹ p = 35 , which is not possible because p is a prime number. 
• Thus, we can say that q≠23 
Thus, the correct answer is  Option C.

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Is the prime number q equal to 23? (1) r = 2p + 1, where p is prime

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