Quotient-Remainder Theorem

Theorem

Given any integer $n$ and positive integer $d$, there exist unique integers $q$ and $r$ such that $n=dq+r$ and $0\le r<d$ $\frac{n}{d}=q$ , with remainder $r$

Example Use

let: $T_0=n\in \mathbb{Z}:n=3k\text{, for some integer }k,$ $T_1=n\in \mathbb{Z}:n=3k+1\text{, for some integer }k,$ $T_2=n\in \mathbb{Z}:n=3k+2\text{, for some integer }k$ Is $\{T_0,T_1,T_2\}$ a partition of $\mathbb{Z}$?

Yes. By the quotient-remainder theorem, every integer $n$ can be written in exactly one of the three forms: $n=3k$, $n=3k+1$, or $n=3k+2$ for some integer $k$. This means that $T_0,T_1,T_2$ are mutually disjoint and $\mathbb{Z}=T_0\cup T_1\cup T_2$