show that any positive odd integer is of the form 6q+1, 6q+3 or 6q+5, where q is any positive integer.
The positive integers +1, +3 and +5 are not of that form, unless q can also be zero.
The difference between 6q + 5 and 6q' +1, if q' = q+1 , is
[6(q+1) + 1]-[6q +5] = 2
That means that by increasing q by 1 when you get to 6q+5, the next number will be 2 greater, so that all odd numbers can be created. All one has to do is pick the next integer q when you get to 6q +5, and start over at 6q' +1
To show that any positive odd integer is of the form 6q+1, 6q+3, or 6q+5, where q is any positive integer, we can use the concept of modular arithmetic.
1. First, we need to understand that any odd integer can be expressed in the form of 2k+1, where k is an integer.
2. Now, let's consider the remainders when dividing the positive integers by 6. The possible remainders are 0, 1, 2, 3, 4, and 5.
3. We can start by considering the remainder 0. If an odd integer gives a remainder of 0 when divided by 6, then it can be expressed as 6q, where q is an integer. However, this form does not satisfy the conditions we are trying to prove.
4. Next, let's consider the remainder 1. If an odd integer gives a remainder of 1 when divided by 6, then it can be written as 6q+1, where q is an integer. This form holds true for all odd integers that have a remainder of 1 when divided by 6.
5. Moving on to the remainder 2. If an odd integer gives a remainder of 2 when divided by 6, then it can be written as 6q+2 = 2(3q+1) = 2k, where k = 3q+1. However, 2k is even, not odd. So, there are no odd integers in this form.
6. Next, let's consider the remainder 3. If an odd integer gives a remainder of 3 when divided by 6, then it can be written as 6q+3, where q is an integer. This form holds true for all odd integers that have a remainder of 3 when divided by 6.
7. Moving on to the remainder 4. If an odd integer gives a remainder of 4 when divided by 6, then it can be written as 6q+4 = 2(3q+2) = 2k, where k = 3q+2. Again, 2k is even, not odd. So, there are no odd integers in this form.
8. Finally, let's consider the remainder 5. If an odd integer gives a remainder of 5 when divided by 6, then it can be written as 6q+5, where q is an integer. This form holds true for all odd integers that have a remainder of 5 when divided by 6.
Therefore, we have shown that any positive odd integer is of the form 6q+1, 6q+3, or 6q+5, where q is any positive integer.