Target question: Is the sum of four different positive prime integers an odd integer?
This is a good candidate for rephrasing the target question.

The key concept involved here is that 2 is the only EVEN prime number, and all other prime numbers are ODD

So, if 2 is among the four primes, then the sum = EVEN + ODD + ODD + ODD = ODD
If 2 is NOT among the four primes, then the sum = ODD + ODD + ODD + ODD = EVEN

So, in order for the sum to be ODD, 2 must be among the four primes.
This means we can rephrase the target question as follows:
REPHRASED target question: Is 2 among the four primes?

Statement 1: One of the integers is 11.
Since we have no information about the other three primes, 2 may or may not be among them.
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: One of the integers is 2.
Perfect!
The answer to the REPHRASED target question is YES, 2 is among the four primes
Since we can answer the REPHRASED target question with certainty, statement 2 is SUFFICIENT

Answer: B

Cheers,
Brent

VIDEO ON REPHRASING THE TARGET QUESTION:

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STatement 1: One of the integers is 11.

Prime may be {2, 3, 5, 11} i.e. Sum ODD
Prime may be {3, 5, 7, 11} i.e. Sum EVEN

NOT SUFFICIENT

STatement 2: One of the integers is 2

Now remaining three prime will have to be ODD hence sum of the four primes must be ODD

SUFFICIENT

Answer: Option B
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#1
One of the integers is 11 ( o)
o+o+o+o = even
or o+e+o+o = odd
insufficient
#2One of the integers is 2 ( e)
only possibility
o+e+o+o = odd
sufficient
OPTION B