We want to find the value of X in the list of 7 numbers mentioned in this question.
Statement 1: The average (arithmetic mean) of the 7 numbers is 80.
$$\frac{Summation\ FX}{Summation\ f}=80\ where\ f=7\ and\ fx=X+81+73+71+98+73+64\ =\ x+460$$
$$\frac{X+460}{7}=80$$
$$X+460=560$$
$$X=560-460=100$$
Statement 1 is SUFFICIENT.

Statement 2: The range of these 7 numbers is 36.
To find the range, the 7 numbers must be arranged in ascending order, and with this;
Range = highest number - lowest number
The position of X include;
- X can be the highest number i.e, 64, 71, 73, 73, 81, 98, X.
- X can be the lowest number i.e, X, 64, 71, 73, 73, 81, 98.
- X can neither be the highest nor lowest i.e 64, 71, X, 73, 73, 81, 98.
If X is neither the highest nor lowest, then we cannot use the range to find the value of X.
However, if X is the highest number, then,
Range = X - 64
36 = X - 64
X = 36 + 64 = 100

Also, if X is the lowest number,
Range = 98 - X
36 = 98 - X
X = 98 - 36 = 62
These are the two different and possible values of X in this statement, hence, statement 2 is NOT SUFFICIENT.

Since statement 1 alone is SUFFICIENT, the correct answer is option A. $$$$