Question

show that any positive integer is in the form of 6q , 6q+2 or 6q+4 for q is some integer

Answer 1

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.