x^2 + y^2 = (x+y)^2-2xy = (x-y)^2 + 2xy

If (x+y) = 1 (mod 5) then (x+y)^2 = 4 (mod 5)

since we do not know what is 2xy equal in (mod 5) so we can not know whether (x+y)^2-2xy is equal to 0 or not. INSUFF

If (x-y) = 3 (mod 5) then (x-y)^2 = 9 = 4 (mod 5)

since we do not know what is 2xy equal in (mod 5) so we can not know whether (x-y)^2+2xy is equal to 0 or not. INSUFF

Gathering both datas together =>

(x+y) = 1 mod 5

(x-y) = 3 mod 5

(x+y)^2 + (x-y)^2 = 2(x^2+y^2)

So

(x+y)^2 + (x-y)^2 = 1 + 9 = 10 = 0 (mod 5)

since 2(x^2+y^2) = 10 = 0 (mod 5)

x^2 + y^2 = 0 in module 5.

That is to say, x^2 + y^2 is divisible by 5

Both are needed

Answer is C