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​