Question

Write whether the square of any positive integer can be of the form 3m+2 for some integer where MI's a natural number.justify your answer

Answer 1

Square of any positive integer cannot be of the form 3m+2.

Answer 2

No there will not be any positive integer.

Step-by-step explanation:

Let us assume that square of any positive integer can be of the form 3m+2  

By Euclids division algorithm,

Any positive integer a can be expressed as bq+r

Where q is the quotient and r is remainder.

$a=bq+r$

Let b = 3 (As we need to prove the integer pertaining to 3m+2) and 0 ≤ r <b

a=3q+0; a=3q+1; a=3q+2 …….are positive integers

Consider square of each of integers ….

$(3 q)^{2}=9 q^{2}$

$(3 q+1)^{2}=\left(9 q^{2}+6 q\right)+1=3\left(3 q^{2}+2 q\right)+1$

$(3 q+2)^{2}=\left(9 q^{2}+12 q\right)+4=3\left(3 q^{2}+4 q\right)+4$

From the above equations, we can find that square of any positive integer cannot be of the form 3m+2.