Question

Show that the square of any positive integer can be in the form of 6q+1 or 6q for any integer "m".

Answer 1

Answer:

Step-by-step explanation:

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.Read more on Sarthaks.com - https://www.sarthaks.com/877920/show-that-the-square-of-an-odd-positive-integer-can-be-of-the-form-6q-1-or-6q-3-for-some-integer