How any odd positive integer can be 6q+1,6q+3,6q+5? Why it can't be 2q+1,2q+3,2q+5?
Answers
Odd positive integers can be represented as 6q+1, 6q+3, 6q+5 and we can also represent them as 2q+1, 2q+3, 2q+5 (where q is a non-negative integer)
But considering the fact that every odd number can be represented only in a unique way using the combination 6q+1, 6q+3, 6q+5 makes it more suitable.
What I'm trying to tell is, for eg. consider the odd number 5
Using the combination 2q+1, 2q+3, 2q+5, we can represent 5 in three different ways:-
By substituting q=2 in 2q+1
By substituting q=1 in 2q+3
By substituting q=0 in 2q+5
Similarly, every odd number (except 1,3) can be represented in three different ways, or in other words, substituting values for q will lead to repetition of the same odd number thrice.
But consider 6q+1, 6q+3, 6q+5
Every odd number can be represented by only a single way using this combination.
Consider 5 itself.
We will get 5 by substituting 0 in 6q+5
But no non-negative integer value of q will give 5 in any other way.
This is why I consider 6q+1, 6q+3, 6q+5 as a more apt representation of odd numbers.