If x and y are integer, what is the remainder when x^2 + y^2

Hi, this is my first post on this forum. I was little confused with the way I solved the problem and I don't know where did I go wrong!!

Statement 1 & 2 are insufficient individually..ok

When I combined the two statements, I multiplied the dividends and the remainders:

Dividends: (x-y) * (x+y) = x^2 - y^2
Remainders: 1 * 2 = 2

As we know that the divisor in this case is the same (i.e. 5) I assume we can multiply the different dividends and the remainders. In this way we can have the remainder (which is product of the two individual remainders) when x^2 - y^2 is divided by 5.

Once again: When (x^2 - y^2) / 5 the remainder is 2...

We can rewrite this as 5q + 2 = x^2 - y^2

Now I plugged in different values for q like 1,2,3,4

When q = 1, 5(1) + 2 = 7 -----> x^2 - y^2 = 7
When q = 2, 5(2) + 2 = 12 -----> x^2 - y^2 = 12
When q = 3, 5(3) + 2 = 17 -----> x^2 - y^2 = 17
When q = 4, 5(4) + 2 = 22 -----> x^2 - y^2 = 22

Now we can assume different values of x and y so that our total equals 7,12,17 and 22.

For the first case ( when x^2 - y^2 = 7) we can assume x^2=10 and y^2= 3 so when x^2 - y^2 = 7, then x^2 + y^2 = 10. In this case remainder will be 0 when x^2 + y^2 is divided by 5

For the second case ( when x^2 - y^2 = 12) we can assume x^2=15 and y^2= 3 so when x^2 - y^2 = 12, then x^2 + y^2 = 18. In this case remainder will be 3 when x^2 + y^2 is divided by 5

For the third case ( when x^2 - y^2 = 17) we can assume x^2=21 and y^2= 4 so when x^2 - y^2 = 17, then x^2 + y^2 = 25. In this case remainder will be 0 when x^2 + y^2 is divided by 5

For the fourth case ( when x^2 - y^2 = 22) we can assume x^2=25 and y^2= 3 so when x^2 - y^2 = 22, then x^2 + y^2 = 28. In this case remainder will be 3 when divided by 5

So we are not getting consistent remainders when x^2 + y^2 is divided by 5, aren't Statement 1 and 2 together also insufficient?