**Solution:**

Let a be the positive integer and b = 4.

Then, by Euclid’s algorithm, a = 4m + r for

some integer m ≥ 0 and r = 0, 1, 2, 3 because 0 ≤ r < 4.

So, a = 4m or 4m + 1 or 4m + 2 or 4m + 3.

So, (4m)^{2} = 16m^{2} = 4(4m^{2})

= 4q, where q is some integer.

(4m + 1)^{2} = 16m^{2} + 8m + 1

= 4(4m^{2} + 2m) + 1

= 4q + 1, where q is some integer.

(4m + 2)^{2} = 16m^{2} + 16m + 4

= 4(4m^{2} + 4m + 1)

= 4q, where q is some integer.

(4m + 3)^{2} = 16m^{2} + 24m + 9

= 4(4m^{2} + 6m + 2) + 1

= 4q + 1, where q is some integer.

**Hence, The square of any positive integer is either of the form 4q or 4q + 1, where q is some integer.**