By Euclid’s division algorithm, By Euclid’s Lemma, b = a × q + r, 0 ≤ r < a

Here, b is any positive integer.

Let a be an arbitrary positive integer. Then corresponding to the positive integers a and 5, there exist non-negative integers m and r such that

a = 5m + r, where 0 ≤ r < 5

Squaring both the sides using (a + b)² = a² + 2ab + b²

⇒ a² = (25m² + r² + 10mr)

⇒ a² = 5(5m² + 2mr) + r²

Where 0 ≤ r < 5

Case I When r = 0 we get

a² = 5(5m²)

⇒ a² = 5q

where q = 5m² is an integer.

Case II When r = 1 we get

a² = 5(5m² + 2m) + 1

⇒ a² = 5q + 1

where q = (5m² + 2m) is an integer.

Case III When r = 2 we get

⇒ a² = 5(5m² + 4m) + 4

⇒ a² = 5q + 4

Where q = (5m² + 4m) is an integer.

Case IV When r = 3 we get

⇒ a² = 5(5m² + 6m) + 9 = 5 (5m² + 6m) + 5 + 4

⇒ a² = 5(5m² + 6m + 1) + 4 = 5q + 4

where, q = (5m² + 6m + 1) is an integer.

Case V when r = 4 we get

⇒ a² = 5(5m² + 8m) + 16 = 5 (5m² + 8m) + 15 + 1

⇒ a² = 5(5m² + 8m + 3) + 1 = 5q + 1

where, q = (5m² + 8m + 3) is an integer.

Hence, the square of any positive integer cannot be of the form 5q + 2 or 5q + 3 for any integer q.