1.

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

Answer»

We know that any positive integer can be of the form 6m, 6m + 1, 6m + 2, 6m + 3, 6m + 4 or 6m + 5, for some integer m. 

Thus, an odd positive integer can be of the form 6m + 1, 6m + 3, or 6m + 5 

Thus we have: 

(6 m +1)2 = 36 m2 + 12 m + 1 = 6 (6 m2 + 2 m) + 1 = 6 q + 1, q is an integer 

(6 m + 3)2 = 36 m2 + 36 m + 9 = 6 (6 m2 + 6 m + 1) + 3 = 6 q + 3, q is an integer 

(6 m + 5)2 = 36 m2 + 60 m + 25 = 6 (6 m2 + 10 m + 4) + 1 = 6 q + 1, q is an integer. 

Thus, the square of an odd positive integer can be of the form 6q + 1 or 6q + 3.



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