Is the average of X consecutive integers odd?

If the number of consecutive integers is even the average will be a fraction.
If the number of consecutive integers is odd the average will be an integer.

St1: First number is odd --> We do not know about the number of terms. Insufficient

St2: Sum of the numbers is odd --> Let n be the number of even integers and n' be the number of odd integers
n(Even) + n'(Odd) = Odd
Possible cases: n is even and n' is odd --> Number of terms = n + n' = odd --> Average is an integer. --> Average may be odd or even
n is odd and n' is odd --> n + n' = even --> Average is not an integer --> Average is not odd
Not Sufficient

Combining St1 and St2: First number is odd and sum is odd --> Odd + Even + Odd + Even + Odd --> Sum is odd and average is odd
Odd + Even --> Sum is odd, but average is not odd.
Not Sufficient

Answer: E