Question

Prove that square of every positive iteyer is of the form 3p, 3p + 1​

Answer 1

Let

′

a

′

be any positive integer and b=3,

Then, by division algorithm a=3q+r,

for some integer q⩾0 and 0⩽r⩽3.

So, a=3q,3q+1,3q+2.

Then, the square of positive integer,

a=3q

a

2

=(3q)

2

=9q

2

=3(3q

2

)

=3p

(Where p=3q

2

)

a=3q+1

a

2

=(3q+1)

2

=9q

2

+6q+1

=3(3q

2

+2q)+1

=3p+1

(Where p=3q

2

+2q)

a=3q+2

a

2

=(3q+2)

2

=9q

2

+12q+4

=3(3q

2

+4q+1)+1

=3p+1

(Where p=3q

2

+4q+1)

Since p is some positive integer.

∴ The square of any positive integer is of the form 3p or 3p+1.