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1) 2x+2 is cube of an +ve integer. Only 3 satisfies this condition.

I think sufficient.

2) Avg of x consecutive int is an integer.

x = 3, 5, 7 ...... odd primes INSUFF.
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"To dream anything that you want to dream, that is the beauty of the human mind. To do anything that you want to do, that is the strength of the human will. To trust yourself, to test your limits, that is the courage to succeed."

Somehow, my world of primes stops after 2,3,5,7,11, 13...
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"To dream anything that you want to dream, that is the beauty of the human mind. To do anything that you want to do, that is the strength of the human will. To trust yourself, to test your limits, that is the courage to succeed."

(2) The average of any x consecutive integers is an integer

This is an interesting question. I'll just add my approach for fun.

(1) 2x+2 = 2(x+1) is cube of a positive integer, meaning x+1 is multiples of 4 and cube of other integers. (meaning x is odd too.)
x=4k^3-1
So if k=1, x=3; k=2, x=31; ...etc.

(2) It says x is odd. If x is odd then the average of any x consecutive integers is always an integer.

Combined (2) does not add any new info to (1).

That's why I'd choose E.
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keep on seeking, and you will find;
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