Question = is positive integer x a prime number?
$$Statement\ 1\ =\ When\ \frac{x}{4}\ remainder\ =3$$
From the statement above x could be = 3,7,11,15,19,23,27,31,35,39,43,and 47.
Prime numbers are supposed to be divisible by itself and one (1) but we have four (4) of old numbers in statement 1, hence statement 1 is not sufficient.

Statement 2 = x is not divisible by 3 or 5.
x = 2,4,8,11 etc.......
Unfortunately we already have even numbers, hence x is not sufficient.
Statement 1 and 2 together
Stating the value of x from statement 1
x= 8,7,11,15,19,23,27,31,35,39,43,47 and applying the condition in statement 2
x= 7,11,19,23,31,43,47,(all prime)
Thus both statement together are sufficient.
Final answer is option C.