Strayer Elementary Number Theory 1. 3. 50.

Sad to say this one stumped me. I will explain the method I tried and show the notes detailing what I tried and why it failed. I had to look it up online. Below the notes is the actual proof. Thanks to Math Stack Exchange for this solution (I detailed it out, of course).

Here are my notes. I tried to complete this by showing that for any odd n, n can be written as n=n-2+2, where of course 2,n-2>1 and (2,n-2)=1. This of course is correct and made it to the final solution. What stumped me was the case where n is even. Of course we know just by looking at the integer line that if n is even and greater than 6, there exists some prime 1<p<n-1 such that (n,p)=1. My plan was to show n=n-p+p and n-p,p>1 and (n-p,p)=1. The problem is, showing such a trivial thing as the existence of this p is actually very hard (if not impossible) with just such elementary techniques as divisibility and the greatest common divisor. I tried using sets. I tried using contradiction. I tried using induction. All failed... You could do this more easily if you have Euler's Totient Function, for example... but this technique is not introduced until later. So using it in Ch. 1 would be cheating. So I got stumped when n is even and abandoned ship. Again, thanks to MSE for the solution that is here.

Here is the solution given by MSE, although filled out and given with a lemma.