Is integer r prime? (1) r + 1 is prime (2) r is greater than 10

Here is my solution to this one -->
Given data -->
r is an integer.
WE need to check if r is prime or not.

Statement 1 -->
r+1 is prime.
r+1=3 r=2 => Yes, r is prime
r+1=5 => r=4 => No, r is not prime .

Statement 2-->
No clue of r
Not sufficient

Combing the two statements we can say that r+1 is a prime number greater than 10.
hence r=> even number >2
Thus r can never be a prime number.

Hence C

1) "r + 1 is prime"
r could = 2 with 3 being a prime number that makes this statement true, or r could = 5 with 4 NOT being a prime number also making this statement true.
These two scenarios create both a yes and no answer for the question while satisfying the conditions of the statement
Insufficient

2) "r is greater than 10"
This statement, like statement 1 can create both a yes and no answer for the question while still satisfying the conditions of the statement (11, 12, 13, 14, etc.)
Insufficient

Statements 1 and 2 combined
From the analysis of statement 1, the only way r could equal a prime number based on the limits of the statements, is if r equals 2. Statement 2 states that r is greater than 10 which eliminates the possibility of r being a prime number.

Answer: C

Is r=prime?
statement 1:
r+1=prime
---> 12+1=13 (prime)
so, r is NOT prime.
r+1=prime
---> 2+1=3 (prime)
So, r IS prime.
Statement 1 is INSUFFICIENT

Statement 2:
r>10
Here, r may be=11 (which IS prime)
again, r may be= 12 (which is NOT prime)
Statement 2 is INSUFFICIENT

Statement 1+ statement 2:
here, r+1 must be prime where r is greater than 10
if r+1 is prime, then r must be even. So, r is not prime because EVEN integer ONLY 2 is prime in the real world. So, r is NOT prime.
---> Sufficient
C is the correct choice to me...

We need to determine whether r is prime.

Statement One Alone:

r + 1 is prime.

Statement one alone is not sufficient to answer the question. For example,

If r = 6, then the answer is no. If r = 2, then the answer is yes.

Statement Two Alone:

r is greater than 10.

Statement two alone is not sufficient to answer the question. For example,

If r = 12, then the answer is no. If r = 11, then the answer is yes.

Statements One and Two Together:

From statements one and two, we see that r is greater than 10 and r + 1 is prime. Since all prime numbers greater than 2 are odd, we see that r + 1 must be odd. In that case, r must be even. However, since r > 10, it can’t be a prime. So the answer is no.

Answer: C

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