TITLE: What does this number theory statement mean?
QUESTION [6 upvotes]: I recently started studying number theory by myself and I am reading a book about number theory. There is one thing that I don't understand, the statement below:

If $a,b \in \mathbb{Z}$, then there is a $d \in \mathbb{Z}$ such that $(a,b)=(d)$.

I understand everything but the $(a,b)=(d)$. I know it has to do something with set theory probably, but what? Specifically why d is in parentheses and a pair is equal to a single variable?

REPLY [4 votes]: $(a,b)$ means (in simple words) everything you get by multiplying and adding whatever is in $\mathbb{Z}$. More precisely, it is the ideal generated by $a$ and $b$. Similarly, $(d)$ is generated by $d$ single handedly, i.e., its multiples.
Two elements generate an ideal, which is "as fine as" their gcd. Indeed, this is what gcd means: it is the "finer mesh" that covers their union. The gcd is the "common nature" of them, which may also give this "joined mesh". For example, adding and multiplying whatever by 6 and 10, then you get all multiples of 2. Here, gcd(6,10)=2. This is why gcd is also notated as (6,10).
If you want a proof of this, you probably may find it in the section you ar reading, no matter what book you are referring to. Or you may want to show it yourself.