Question: I have scoured my textbook for the concept alluded to in the title of this thread; however, my textbook has failed, in that it provides no such information.

Does anyone know of some resources for this concept?

EDIT:

For instance, let me post a problem I am working on:

An instructor has given a short test consisting of two parts. For a randomly selected student, let $X=$ the number of points earned on the first part and $Y=$ the number of points earned on the second part. Suppose that the joint pmf of X and Y is given in the accompanying table.

p(x,y) 0 | 5 | 10 | 15

0 | 0.02| 0.06| 0.02| 0.10

5 | 0.04| 0.15| 0.20| 0.10

10| 0.01| 0.15| 0.14| 0.01

a) If the score recorded in the grade book is the total number of points earned on two parts, what is the distribution of $X+Y$ and what is the expected recorded score $E(X+Y)$

b) If the maximum of the two scored is recorded, what is the distribution of the maximum score $Max(X,Y)$ ad the expected maximum score $E(Max(X,Y))$

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Now that I re-look at this question, I doubt whether I can solve any of it. At any rate, how do I find the distribution of $X+Y$?

Answer: 
The set $\rm\,S\,$ of positive integers of the form $\rm\:ma+nb\:$ is closed under positive subtraction, i.e. if $\rm\: j,k\in S\:$ then $\rm\:j > k\:\Rightarrow\:j-k \in S.\:$ So, by a simple [fundamental lemma,](https://math.stackexchange.com/a/363623/23500) the least positive element $\rm\:d\in S\:$ divides every element of $\rm\:S.\:$ Thus $\rm\:a,b\in S\:\Rightarrow\: d\mid a,b,\:$ i.e. $\rm\:d\:$ is a *common* divisor of $\rm\:a,b.\:$ Conversely, $\rm\:c\mid a,b\:\Rightarrow\: c\mid d = ma + nb\:\Rightarrow\:c\le d,\:$ so $\rm\:d\:$ is the *greatest* common divisor (i.e. any common divisor $\rm\:d\:$ of $\rm\:a,b\:$ having linear form $\rm\:d = ma+nb\:$ is necessarily *greatest*). 

Hence we see that Bezout's identity for the gcd is just a special case of said fundamental lemma. This lemma has widespread applications in elementary number theory. The key innate structure is clarified when one studies university algebra: ideals are *principal* in Euclidean domains (and ideal-theoretic structure is hidden everywhere in elementary number theory).

Remark: it is is easy to verify the claim that $\rm\,S\,$ is closed under (positive) subtraction:

$$\rm\begin{eqnarray} j\, =\, ma+nb\in S\\ \rm k\, =\, \hat m a + \hat n b\in S\end{eqnarray}\bigg\rbrace,\ \ \, j>k\ \ \Rightarrow\ \ j-k\, =\, (m-\hat m)\, a + (n-\hat n)\,b \in S$$
From [Bezout's identity](http://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity) we have that $\gcd(a,b)=sa+rb$ for some $r,s\in\mathbb Z$.  

Also $\gcd(a,b)\mid a , \ \gcd(a,b)\mid b$ and therefore for any $n,m\in\mathbb Z$ with $ma+nb>0\Rightarrow ma+nb=k\cdot\gcd(a,b)$ for some $k\in\mathbb N$.  

Since $sa+rb=1\cdot\gcd(a,b)$ it follows that $\gcd(a,b)$ is the smallest element of $\lbrace ma + nb : m, n\in\mathbb{Z}\text{ and }ma+nb>0\rbrace$.