XY + X + Y...How would I solve? Consider two positive even integers less than $15$ (not necessarily distinct). When the sum of these two numbers is added to their product, how many different possible values may result?
I was thinking that it would be 15^2. Then, I realized that some of the values where numbers x and y are tiny are the same. Can someone find the answer, and explain it to me?
 A: Fleshing out my comment and suggestions from others, here is are two techniques to solve the problem.
If you are like me, your first instinct is usually brute force, which is usually not a good thing. But in this case it is fine. I wrote the following Mathematica code:
Length[DeleteDuplicates[Flatten[Table[x y + x + y, {x, 2, 14, 2}, {y, 2, 14, 2}]]]]

Which returns the number 27, that being the answer to your question.
A trickier technique suggested by @arts is to notice that the number of unique integers of the form $XY+X+Y$ is the same as the number of integers of the form $$XY+X+Y+1=(X+1)(Y+1).$$ Call the set of all such elements $S$. If a number $Z$ is in $S$, then $Z=(X+1)(Y+1)$, with $X+1,Y+1>1$, so it is a composite number. Next, since $X$ and $Y$ must be even, we know that $X+1$ and $Y+1$ are odd, so $Z$ is also odd. Finally, both $X+1$ and $Y+1$ must be less than or equal to 15. Therefore, the set $S$ consists of all odd composite numbers which can be written as the product of two numbers less than or equal to 15. Now, this is still not a very elegant solution because you have to go through all the odd composite numbers and decide which can be written as a product of two numbers less than or equal to 15, and which cannot. But You will also arrive at your answer this way by counting the number of elements in $S$.
