What does $\mathbb{Z}_{7429}$ mean? What exactly does $\mathbb{Z}_{7429}$ mean? Is it the set of all integers up to and including 7429?
 A: I choose to answer with a more complete explanation because I once also thought that $\mathbb{Z}_n = \{ -(n-1), ..., 0, ..., n-1\}$
To fully understand this group, you must understand the equivalence relation congruence modulo $n$. We say that $(a,b) \in R$ or using an equivalent notation $aRb$ iff $a = b+nk$ $,k \in \mathbb{Z}$.
As an example: congruence modulo $2$.
$(2,4)$ is an element of this relation because $2 = 4 + 2k$, namely $k=-1$.
Now we define the equivalence class of the relation congruence modulo $n$ as following:
$$\bar{a} = \{b \mid aRb \}$$
going back to our example of congruence modulo $2$, this would mean that, for example:
$$\bar{2}= \{b\mid2Rb\} = \{b\mid2=b+2k \} = \{0,2,-2,...\} = \bar{4} = ...$$
Now we can finally define $\mathbb{Z}_n$:
$$\mathbb{Z}_n = \{\bar{0},...,\overline{n-1}\}$$
Hopefully this makes things a bit clearer for you.
A: For the Additive Group of Integers Modulo $m$. which is a cyclic group, the notation $\mathbb{Z}_m$ is often used.
This is the group where $7429\equiv0$ with the usual addition.
A: Another notation for $\mathbb{Z}/7429\mathbb{Z}.$
A: $\mathbb{Z}_{7429}$ is the set of all integers modulo $7429$.
