Question

Show that any positive intergers is of the form 3 m
or (3m + 1) or (3m + 2) or some interger m.​

Answer 1

By Euclid’s division algorithm a = bq + r, where 0 ≤ r ≤ b Put b = 3 a = 3q + r, where 0 ≤ r ≤ 3 If r = 0, then a = 3q If r = 1, then a = 3q + 1 If r = 2, then a = 3q + 2 Now, (3q)2 = 9q2 = 3 × 3q2 = 3m, where m is some integer (3q + 1)2 = (3q)2 + 2(3q)(1) + (1)2 = 9q2 + 6q + 1 = 3(3q2 + 2q) + 1 = 3m + 1, where m is some integer (3q + 2)2 = (3q)2 + 2(3q)(2) + (2)2 = 9q2 + 12q + 4 = 9q2 + 12q + 4 = 3(3q2 + 4q + 1) + 1 = 3m + 1, where m is some integer Hence the square of any positive integer is of the form 3m, or 3m +1 But not of the form 3m + 2Read more on Sarthaks.com - https://www.sarthaks.com/26083/prove-that-the-square-of-any-positive-integer-is-of-the-form-3m-or-3m-1-but-not-of-the-form-3m-2