Question

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

Answer 1

By Euclid’s division Algorithm

a = bm + r, where 0 ≤ r ≤ b

Put b = 4

a = 4m + r, where 0 ≤ r ≤ 4

If r = 0, then a = 4m

If r = 1, then a = 4m + 1

If r = 2, then a = 4m + 2

If r = 3, then a = 4m + 3

Now, (4m)² = 16m²

= 4 × 4m²

= 4q where q is some integer

(4m + 1)² = (4m)² + 2(4m)(1) + (1)2

= 16m² + 8m + 1

= 4(4m² + 2m) + 1

= 4q + 1 where q is some integer

(4m + 2)² = (4m)² + 2(4m)(2)+(2)²

= 16m² + 24m + 9

= 16m² + 24m + 8 + 1

= 4(4m² + 6m + 2) + 1

= 4q + 1, where q is some integer

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