Question

Prove that the square of any positive integer is of the form 5q, 5q + 1, 5q + 4 for some integer q.

Answer 1

By Euclid’s division algorithm

a = bm + r, where 0 ≤ r ≤ b

Put b = 5

a = 5m + r, where 0 ≤ r ≤ 4

If r = 0, then a = 5m

If r = 1, then a = 5m + 1

If r = 2, then a = 5m + 2

If r = 3, then a = 5m + 3

If r = 4, then a = 5m + 4

Now, (5𝑚)² = 25m²

= 5(5m²)

= 5q where q is some integer

(5m + 1)² = (5m)² + 2(5m)(1) + (1)²

= 25m² + 10m + 1

= 5(5m² + 2m) + 1

= 5q + 1 where q is some integer

(5m + 1)² = (5m)² + 2(5m)(1)(1)²

= 25m² + 10m + 1

= 5(5m² + 2m) + 1

= 5q + 1 where q is some integer

= (5m + 2)² = (5m)² + 2(5m)(2) + (2)²

= 25m² + 20m + 4

= 5(5m² + 4m) + 4

= 5q + 4, where q is some integer

= (5m + 3)² = (5m)² + 2(5m)(3) + (3)²

= 25m² + 30m + 9

= 25m² + 30m + 5 + 4

= 5(5m² + 6m + 1) + 4

= 5q + 1, where q is some integer

= (5m + 4)² = (5m)² + 2(5m)(4) + (4)²

= 25m² + 40m + 16

= 25m² + 40m + 15 + 1

= 5(5m²) + 2(5m)(4) + (4)²

= 5q + 1, where q is some integer

Hence, the square of any positive integer is of the form 5q or 5q + 1, 5q + 4 for some integer q.