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

Any number is of the following forms:
$3x + 1 \\ 3x + 2 \\ 3x$
Now squaring all these, we get
${(3x + 1)}^{2} = 9 {x}^{2} + 1 + 6x = 3x(2 + 3x) + 1 = 3m + 1 \\ {(3x + 2)}^{2} = 9 {x}^{2} + 4 + 12x = 9 {x}^{2} + 3 + 12x + 1 = 3(3 {x}^{2} + 1 + 4x) + 1 = 3m + 1 \\ {(3x)}^{2} = 9 {x}^{2} = 3(3 {x}^{2} ) = 3m$
Since none of these is of the form 3m+2, there is no perfect square of the form 3m+2.