What does notation $(a_1, a_2,\cdots, a_n)$ mean in book "The Classical Introduction to Modern Number Theory"? I am reading the book on number theory and I have problems understanding this definition:

DEFINITION: Let $a_1, a_2, \cdots, a_n\in\mathbb{Z}$; we define $$\left(a_1, a_2, \cdots, a_n\right):=\left\{a_1x_1+\cdots+a_nx_n\,:\,x_1,\cdots, x_n\in\mathbb{Z}\right\}$$ 

If I got it right, then every set $A=(a,b)$ contains all integer numbers. Let me make it a little bit more clear.
$$A=(5,7)\\
0*5+0*7=0\\
(-4)*5+3*7=1\\
(-1)*5+1*7=2\\
(-5)*5+4*7=3$$ et cetera...
I can do same thing but swap the $+$ and $-$ signs in coefficients, and get all the negative integers too. And I can do it with every other pair of numbers, and triplet and so on.
So my question is, did I get this right?

It is correct that $(5,7)$ is the set of all integers. 
However, this does not make the definition meaningless. 
For example, if all $a_i$ are even, clearly each element in $(a_1, \dots, a_n)$ will be even. 
So the set $(a_1, \dots, a_n)$ is not always the set of all integers. 
Presumably, the book will proceed to show that $(a_1, \dots, a_n)= d \mathbb{Z}$ for some (non-negative) integer $d$ and this $d$ is a GCD of $a_1, \dots, a_n$. 
