Question

Is this possible to have the square of all the positive integers of the form 3m + 2, where m is a natural number. Justify your answer.
(Class 10 Maths Sample Question Paper)

Answer 1

It is not possible at all to have the squares of all the positive integers of the form 3m + 2.

This can be explained by using Euclid's Lemma,a= bq+r, 0≤ r <b, where b is an integer.

Let b= 3 , then a= 3q +r  (0≤ r <3)

here r =0,1,2,

For , a= 3q

a= (3q)² (on squaring)

a= 9q² = 3(3q²)

a= 3m , [ where m = 3q²]

For ,a= (3q+1)²

a= 9q² + 6q +1

[ (a+b)² = a² + 2ab +b²]

a= 3( 3q²+2q) +1

a= 3m +1,  [ where m = 3q²+ 2q]

For ,a= (3q+2)²

a= 9q² + 12q +4

a= 9q² + 12q +3 +1

[ (a+b)² = a² + 2ab +b²]

a= 3( 3q²+4q+1) +1

a= 3m +1,  [ where m = 3q²+ 4q +1]

It is clear that the square of any positive integer cannot be of the form 3m+ 2.

HOPE THIS WILL HELP YOU..