Question

prove that if a positive integer is of the form 6q+5 ,then it is of the form 3q+2 for some integer q ,but not conversely.

Answer 1

Since any positive integer is in form of 3m, 3m + 1 and 3m + 2.

Case 1 : Taking q = 3m

⇒ x = 6q + 5

⇒ x = 6(3m) + 5

⇒ x = 18m + 5

⇒ x = 3(6m + 1) + 2 ⇒ x = 3q + 2

Case 2 : Taking q = 3m + 1

⇒ x = 6(3q + 1) + 5

⇒ x = 18m + 6 + 3 + 2

⇒ x = 3(6m + 3) + 2 ⇒ x = 3q + 2

Case 3 : Taking q = 3m + 2

⇒ x = 6(3m + 2) + 5

⇒ x = 18m + 12 + 3 + 2

⇒ x = 3(6m + 5) + 2 ⇒ x = 3q + 2

Conversely,

Every positive integer can be in form of 6q, 6q + 1 and 6q + 2.

⇒ x = 3(6n) + 2 ⇒ x = 6(3n) + 2

⇒ x = 6q + 2

6q + 2 ≠ 6q + 5

Therefore, if a positive integer is of the form 6q+5 ,then it is of the form 3q+2 for some integer q ,but not conversely.

Q.E D

Answer 2

Answer:

proof

let  x be positive integer is of the form 6q+5

x = 6q +5   here q is 0,1,2,3.......

x =6q +3+2

x =3[2q+1] +2

x = 3Q +2   here Q is 1,3,5,7...............

x = 3q +2   here q  is 1,3,5,7...............

hence proved  

converse

if x =14

x = 3[4] +2  is in the form    x = 3q +2  

it can not be written in the form of   x = 6q +5  

so converse is not true  

note

we proved  

if a positive number is in the form 6q +5

can be written as 3q +2   for some q

but converse part is not true

so we use an example to dosprove  

in general if  x = 3q +2 where q is even

                  x =3[2k] +2

               x =6k +2   and it can not be written  as x =6q +5