Question

whether the square of any positive integer can be of the form of 3n+2 where N is natural number​

Answer 1

Step-by-step explanation:

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 form of 3m + 2