Question

Show that any positive odd integer is of the form 3q or 3q+1 or 3q+2 for some int

Answer 1

Hi Mate !!


Let a be any odd positive integer which when divided by 3 gives q as quotient and r as remainder .

According to Euclid's division lemma

a = bq + r

and 0 ≤ r < b

Here , b = 3

So,

a = 3q + r

0 ≤ r < b

r = 0 , 1 , 2

So,

a = 3q + 0

=> a = 3q

a = 3q + 1

and a = 3q + 2 !!

It can be written as above !!

Answer 2

Answer .

Given ; 3q ,3q +1 , 3q + 2

Show that ; Odd integer .

Sol ; According to Euclid division lemma ;

b = aq + r

0 < r < a

a = 3

r = 0 ,1,2

case 1 ;

r = 0

3q + 0 = 3q

3q is even integer .

Case 2 ;

r = 1

3q + 1

3q + 1 is odd integer .

Case 3 ;

r = 2

3q + 2

3q + 2 is even integer.

Hence proved .