show that any positive integer is of the form 4q, 4q+2, where q is any positive integer.

To show that any positive integer is of the form 4q or 4q+2, where q is any positive integer, we can use the concept of division and remainders.

Let's consider any positive integer, n. When n is divided by 4, we can have four possible remainders: 0, 1, 2, or 3.

Case 1: Remainder is 0
If n divided by 4 gives a remainder of 0, then n is a multiple of 4. In this case, we can say that n is of the form 4q, where q is equal to n divided by 4.

Case 2: Remainder is 1
If n divided by 4 gives a remainder of 1, then we can say that n is one more than a multiple of 4. In other words, n can be expressed as n = 4q + 1, where q is equal to (n-1) divided by 4.

Case 3: Remainder is 2
If n divided by 4 gives a remainder of 2, then we can say that n is two more than a multiple of 4. In other words, n can be expressed as n = 4q + 2, where q is equal to (n-2) divided by 4.

Case 4: Remainder is 3
If n divided by 4 gives a remainder of 3, then we can say that n is three more than a multiple of 4. In other words, n can be expressed as n = 4q + 3, where q is equal to (n-3) divided by 4.

Therefore, any positive integer can be written in the form 4q or 4q+2, where q is a positive integer.