Question

Show that the square of an odd positive integer can be of the form 6q + 1 or 6q + 3 for some integer q.

Answer 1

Given: Let n be an odd positive integer. Any odd integer is of the form 6t + 1, 6t + 3 or 6t + 5 for some integer t.

Case 1: n = 6t + 1n²

= (6t + 1)² 

= 36t² + 12t + 1 

= 6(6t² + 2t) + 1, so n² is of the form 6q + 1.

Case 2: n = 6t + 3n²

= (6t + 3)² 

= 36t² + 36t + 9

= 6(6t² + 6t + 1) + 3, so n² is of the form 6q + 3.

Case 3: n = 6t + 5n²

= (6t + 5)² 

= 36t² + 60t + 25 

= 6(6t² + 10t + 4) + 1, so n² is of the form 6q + 1.

From the three possible residue classes for an odd integer modulo 6, its square is either of the form 6q + 1 or 6q + 3.