Question

show that any positive integer is of the form that 6q.6q+2 or 6q +4 where q is some integer

Answer 1

Answer:
To show any positive even integer is of the form that 6q, 6q + 2 or 6q + 4 where q is some integer

Let ' a ' is any positive integer and b = 6 than we apply Euclid's division lemma , and get

a = 6q + r , So here 0$\le$ r < 6
So r will be 0 , 1 , 2 , 3 , 4 , 5
So At r = 0 , we get
a = 6q
Here at any value of q = 0 , 1 , 2 , 3 , 4 ......... n
a = even number

At r = 1 , we get
a = 6q + 1
Here at any value of q = 0 , 1 , 2 , 3 , 4 ......... n
a = Odd number

At r = 2 , we get
a = 6q + 2
Here at any value of q = 0 , 1 , 2 , 3 , 4 ......... n
a = even number

At r = 3 , we get
a = 6q + 3
Here at any value of q = 0 , 1 , 2 , 3 , 4 ......... n
a = Odd number

At r = 4 , we get
a = 6q + 4
Here at any value of q = 0 , 1 , 2 , 3 , 4 ......... n
a = even number

At r = 5 , we get
a = 6q + 5
Here at any value of q = 0 , 1 , 2 , 3 , 4 ......... n
a = Odd number

Hence any positive even number is in form of 6q , 6q + 2 or 6q + 4 . ( Hence proved )