To show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer, we can use the properties of integers and divisibility.
Let's consider an odd integer, denoted by 'n'.
Since it is an odd integer, we know that it cannot be divisible by 2. Therefore, 'n' can be expressed as 2k + 1, where 'k' is some integer.
Now, we need to show that 'n' can also be represented in the form 6q + 1, 6q + 3, or 6q + 5, where 'q' is some integer.
Let's consider three cases:
Case 1: When n = 6q
If 'n' is divisible by 6, then it can be expressed as 6q, where 'q' is an integer. But this means 'n' is even (since it can be written as 2 * 3q), contradicting the fact that 'n' is an odd integer.
Case 2: When n = 6q + 2
If 'n' is of the form 6q + 2, where 'q' is an integer, then 'n' can be written as 2(3q + 1). This means 'n' is even (since it is divisible by 2), contradicting the fact that 'n' is an odd integer.
Case 3: When n = 6q + 4
If 'n' is of the form 6q + 4, where 'q' is an integer, then 'n' can be written as 2(3q + 2). This means 'n' is even (since it is divisible by 2), contradicting the fact that 'n' is an odd integer.
Now, we are left with only one possible form for 'n':
n = 6q + 1, or n = 6q + 3, or n = 6q + 5
In each of these forms, 'n' can be expressed as an odd integer, and it covers all possible odd integers as 'q' can be any integer (positive, negative, or zero).
Therefore, any positive odd integer is indeed of the form 6q + 1, or 6q + 3, or 6q + 5, where 'q' is some integer.