Question

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

Answer 1

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 = 16m² = 4(4m²)
= 4q, where q is some integer.
(4m + 1)² = 16m² + 8m + 1
= 4(4m² + 2m) + 1
= 4q + 1, where q is some integer.
(4m + 2)² = 16m² + 16m + 4
= 4(4m² + 4m + 1)
= 4q, where q is some integer.

(4m + 3)² = 16m² + 24m + 9
= 4(4m² + 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.