What Proof Strategy to use I have this theorem(see below) that I am trying to prove. However, I am struggling with how to get started; I don't understand what which proof strategy to use like proof by contradiction, if P then Q, ect... Can someone please explain how you can determine which strategy your are suppose to use by looking at this theorem.  
Theorem: The sum of three consecutive natural numbers is divisible by 3. 
 A: Consider a sequence of natural numbers $n - 1$, $n$, and $n + 1$. Since the question wants you to add them up, do so. Check if this gives you a multiple of 3.
The smart choice of $n - 1$, $n$, $n + 1$ is "easier" than the choice of $n$, $n + 1$, $n + 2$ in many cases, and is a useful trick to know
A: Add three consecutive integers $x, x+1$ and $x+2$, and then show that you have $3x+3$, which must be divisible by 3 since $x$ is a whole number.
This method is called deduction.
A proof by contradiction would be to say suppose the sum were not divisible by 3. Then proceed in the same manner to $3x+3$. Now suppose this were not divisible by 3, then 3x would not be divisible by 3, making x not an integer. This contradicts our earlier assumption that $x$ is an integer, so our supposition that the sum were not divisible by $3$ cannot be true.
The strategy I would generally recommend is to begin by constructing the various components of the proposition to be proved.  So I constructed the three consecutive integers in this example.  Note that the three integers I created can represent any 3 integers, that's why I used $x$ and not some specific three integers. Then attempt to place them into the relationship being proposed by the theorem and either try to break the relationship or find a way of demonstrating that it is an inescapable consequence of the assumptions. Show that the theorem must always be true, by deduction. Or find some single counterexample to its negation.
A: The other answers are all correct, but don't directly address the OPs question. He or she is asking for a strategy for questions like this.
In problems where you see a phrase like "$k$ consecutive integers" a good strategy is often to start with "consider the numbers $n, n+1, \ldots ,n+k-1$" or, when $k=2j+1$ is odd, "consider the numbers $n-j, n-j+1, \ldots, 0, \ldots ,n+j$." Then proceed with whatever works - that might be proof by contradiction or something else. In this case just adding them up does the trick.
