There is a set of consecutive even integers. What is the

Two very important properties of standard deviation:

If we add or subtract a constant to each term in a set:
Mean will increase or decrease by the same constant.
SD will not change.

If we increase or decrease each term in a set by the same percent (multiply all terms by the constant):
Mean will increase or decrease by the same percent.
SD will increase or decrease by the same percent.

You can try it yourself:
SD of a set: {1,1,4} will be the same as that of {5,5,8} as second set is obtained by adding 4 to each term of the first set.

That's because Standard Deviation shows how much variation there is from the mean. And when adding or subtracting a constant to each term we are shifting the mean of the set by this constant (mean will increase or decrease by the same constant) but the variation from the mean remains the same as all terms are also shifted by the same constant.

Back to the original question:

There is a set of consecutive even integers. What is the standard deviation of the set?

(1) There are 39 elements in the set --> SD of a set of ANY 39 consecutive even integers will be the same, as any set of 39 consecutive even integers can be obtained by adding constant to another set of 39 consecutive integers. For example: set of 39 consecutive integers {4, 6, 8, ..., 80} can be obtained by adding 4 to each term of another set of 39 consecutive integers: {0, 2, 4, ..., 76}. So we can calculate SD of {0, 2, 4, ..., 76} and we'll know that no matter what our set actually is, its SD will be the same. Sufficient.

(2) The mean of the set is 382 --> knowing mean gives us nothing, we must know the number of terms in the set, as SD of {380, 382, 384} is different from SD of {378, 380, 382, 384, 386}. Not sufficient.

Answer: A.

Hope it's clear.