Explanation:

Let HCF of (x, y) be h. Then we can write x = h * a and y = h * b. Furthermore, note that HCF (a, b) = 1. This is a very important property. One that seems obvious when it is mentioned but a property a number of people overlook.

So, we can write x = 35a; y = 35b

x + y = 1085 => 35(a + b) = 1085. => (a + b) = 31. 
We need to find pairs of co-prime integers that add up to 31. 

Since 31 is prime. All pairs of integers that add up to 31 will be co-prime to each other. 
eg-(1,30 )(2,29) .....
Or, there are totally 15 pairs that satisfy this condition.

The question is "How many pairs of positive integers x, y exist such that HCF of x, y = 35 and sum of x and y = 1085?"

Hence the answer is "15"