Hi j_shreyans,

You might find it easiest to "play around" with this question a bit. Write down a number of possible values for each piece of information and then eliminate possibilities if they DON'T 'match up' with other pieces of information. In this way, the process is similar to TESTing VALUES.

We're told that X is a PRIME number. We're asked for the value of X.

Fact 1: 2X+2 is the CUBE of a positive integer.

Here are the first few positive cubes: 1, 8, 27, 64, 125

Now let's solve for X in each of these scenarios....

2X + 2 = 1
2X = -1 X is negative. Eliminate this option.

2X + 2 = 8
2X = 6
X = 3
3 IS PRIME. X could be 3....

2X + 2 = 27
2X = 25
X = 12.5 This is NOT prime. Eliminate this option.

2X + 2 = 64
2X = 62
X = 31
31 IS PRIME. X could be 31...
Fact 1 is INSUFFICIENT

Fact 2: The average of any X CONSECUTIVE integers is an integer.

This is actually a Number Property rule - it tells us that X MUST be ODD.

Here are some examples.....
2 consecutive integers -- 1, 2 Average = 1.5
3 consecutive integers -- 1, 2, 3 Average = 2
4 consecutive integers -- 1, 2, 3, 4 Average = 2.5
5 consecutive integers -- 1, 2, 3, 4, 5 Average = 3
Etc.

We now know that X is ODD and we know that it's PRIME.
X could be 3, 5, 7, 11,....31
Fact 2 is INSUFFICIENT

Combined (from our prior work) we know that X could be 3 or 31.
Combined, INSUFFICIENT

Final Answer: E

GMAT assassins aren't born, they're made,
Rich