Question

show that any positive even integer is not in form of 6q or 6q+2 or 6q+4 where q is some integer.

Answer 1

Hello.
1st Method :

Let ‘a’ be any positive even integer and ‘b = 6’.

Therefore, a = 6q +r, where 0 ≤ r < 6.

Now, by placing r = 0, we get, a = 6q + 0 = 6q

By placing r = 1, we get, a = 6q +1

By placing, r = 2, we get, a = 6q + 2

By placing, r = 3, we get, a = 6q + 3

By placing, r = 4, we get, a = 6q + 4

By placing, r = 5, we get, a = 6q +5

Thus, a = 6q or, 6q +1 or, 6q + 2 or, 6q + 3 or, 6q + 4 or, 6q +5.

But here, 6q +1, 6q + 3, 6q +5 are the odd integers.

Therefore, 6q or, 6q + 2 or, 6q + 4 are the forms of any positive even integers.

2nd Method :
Let a be any positive integer and b = 2 . Then , by Euclids division Lemma there exist integers q and r such that 
a = 6q + r , where 0<r<2
0<r<2 = 0<r<1  =r = 0 or r = 1 or r = 2 or r = 3 or r = 4
a = 6q or a = 6q + 1 or 6q + 2 or 6q +3 or 6q+ 4
If a = 6q , then a is an even integer . 

Hope it helps.