If a composite number is any positive integer greater than one that is

Question

If a composite number is any positive integer greater than one that is not prime, is integer r a composite number?

(1) r + 1 is prime
(2) r is greater than 10

SaquibHGMATWhiz · Answer

Solution:

We are asked if the positive integer  r  is composite or not. This is a YES-NO question

Statement 1:    r+1  is prime

Let us take 2 cases:

Case 1:   r=2  (prime) and  r+1=3  is also prime 
 Case 2:   r=4  (composite) and  r+1=5  is prime

We are getting two contradicting results thus  statement 1 alone is not sufficient and we can eliminate options A and D

Statement 2:   r is greater than 10

If r = 11, then r is prime 
If r = 12, then r is composite

Thus,  statement 2 alone is also not sufficient and we can eliminate option B

Combining:

Upon combining,  we can be sure that r is composite

Because no two consecutive integers greater than 10 are primes. In fact, the only two consecutive prime integers are 2 and 3

Thus, if we know that  r + 1 is prime and r > 10 ,  we can be sure that r has to be composite

Hence the right answer is    Option C

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