Question

write whether the square of any positive integer can be of the form 3m+2 where m is a natural number justify your answer​

Answer 1

Answer:

No.

Justification:

                         Let a be any positive integer. Then by Euclid’s division lemma, we have a = bq + r, where 0 ≤ r < b For b = 3, we have a = 3q + r, where 0 ≤ r < 3 ...(i) So, The numbers are of the form 3q, 3q + 1 and 3q + 2. So, (3q)2 = 9q2 = 3(3q2) = 3m, where m is a integer. (3q + 1)2 = 9q2 + 6q + 1 = 3(3q2 + 2q) + 1 = 3m + 1, where m is a integer. (3q + 2)2 = 9q2 + 12q + 4, which cannot be expressed in the form 3m + 2. Therefore, Square of any positive integer cannot be expressed in the form 3m + 2.