Use Euclid’s division lemma to show that the square of any positive integer is e

Use Euclid’s division lemma to show that the square of any positive integer is either of form 3m or 3m + 1 for some integer m.

[Hint: Let x be any positive integer then it is of the form 3q, 3q + 1 or 3q + underline 2. Now square each of these and show that they can be rewritten in the form 3m or 3m +1.]

Answer:

Let a be any positive integer and b = 3 Then a = 3q + r for some integer q >= 0 And r = 0 1, 2 because 0 <= r < 3 Therefore, a = 3q or 3q + 1 or 3q + 2 Or, a ^ 2 = (3q) ^ 2 or (3q + 1) ^ 2 or (3q + 2) ^ 2 a ^ 2 = (9q ^ 2) or 9q ^ 2 + 6q + 1 or 9q ^ 2 + 12q + 4 = 3(3q ^ 2) or 3(3q ^ 2 + 2q) + 1 or 3(3q ^ 2 + 4q + 1) + 1 = 3k_{1} or 3k_{2} + 1 or 3k_{3} + 1 Where k_{1} k_{2} and k_{3} are some positive integers Hence, it can be said that the square of any positive integer is either of the form 3m or 3m + 1.