Prove that any positive odd integer is of the form 6x + 1, 6x + 3, or 6x + 5

Sure, here is the proof:

1. Any integer can be written in the form of 6x, 6x + 1, 6x + 2, 6x + 3, 6x + 4, or 6x + 5 where x is an integer. This is because when an integer is divided by 6, the remainder is either 0, 1, 2, 3, 4, or 5.

2. Now, we know that an integer is odd if it is of the form 2n + 1 where n is an integer. This means that the integer leaves a remainder of 1 when divided by 2.

3. Looking at the forms we have from step 1, 6x, 6x + 2, and 6x + 4 are all divisible by 2 and hence are even. So, they cannot be our required form.

4. The remaining forms are 6x + 1, 6x + 3, and 6x + 5. These are not divisible by 2 and hence are odd.

5. Therefore, any positive odd integer is of the form 6x + 1, 6x + 3, or 6x + 5.