Question

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

Answer 1

ANSWER

Let a be the positive integer and b=3.

We know a=bq+r, 0≤r<b

Now, a=3q+r, 0≤r<3

The possibilities of remainder is 0,1, or 2.

Case 1 : When a=3q

a

2

=(3q)

2

=9q

2

=3q×3q=3m where m=3q

2

Case 2 : When a=3q+1

a

2

=(3q+1)

2

=(3q)

2

+(2×3q×1)+(1)

2

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

Case 3: When a=3q+2

a

2

=(3q+2)

2

=(3q)

2

+(2×3q×2)+(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

Hence, from all the above cases, it is clear that square of any positive integer is of the form 3m or 3m+1.