If x and y are two positive integers such that the highest common divisor of these numbers is two, then how many values can x take if x + y = 99, simultaneously?

We are given that the highest common divisor of x and y is 2, which means that both of them are even. 
Next we are asked, for how many values of x can we have x+y=99
Now, 99 is an odd number, to get an odd number by addition of two numbers, one of those numbers must be odd and the other must be even. 
But we know that both x and y are even as they are both divisible by 2.
Therefore, no value of x and y is possible which satisfy both the conditions. 

Hence, Option C is the correct answer.