**Solution:**

Since x and y are odd positive integers, we have

x = 2m + 1 and y = 2n + 1

=> x^{2} + y^{2} = (2m + 1)^{2} + (2n + 1)^{2}

= 4m^{2} + 4m + 1 + 4n^{2} + 4n + 1

= 4(m^{2} + n^{2}) + 4(m + n) + 2

**Hence, x**^{2} + y^{2} is an even number but not divisible by 4.