Question

show that any positive odd integer is of the form 3m , 3m+1,3m+2 ,where m is some integer​

Answer 1

Let a be any positive integer. Then, it is of the form 3q or 3q+1 or, 3q+2

So, we have the following cases:

Case I: When a=3q

In this case, we have

a

2

=(3q)

2

=9q

2

=3q(3q)=3m where m=3q

Case II: When n=3q+1

In this case, we have

a

2

=(3q+1)

2

=9q

2

+6q+1=3q(3q+2)+1=3m+1, where m=q(3q+2)

Case III: When a=3q+2

In this case, we have

a

2

=(3q+2)

2

=9q

2

+12q+4=9q

2

+12q+3+1

3(3q

2

+4q+1)+1=3m+1, where m=3q

2

+4q+1