Addition mod n

The addition table of $\mathbb{Z}/n$, one cell per sum: row $i$, column $j$, holding $i + j \bmod n$. Listed in order it is a circulant, each row the one above shifted by one place. Colored by each sum's additive order, it shows $\mathbb{Z}/n$'s subgroups, one for each divisor of $n$. Listed by Chinese remainders, $\mathbb{Z}/mn$'s table is $\mathbb{Z}/m$'s with a copy of $\mathbb{Z}/n$'s in each cell. Listed by $p$-adic digits, $\mathbb{Z}/p^k$'s nests: $p \times p$ blocks of $\mathbb{Z}/p^{k-1}$'s, down to single cells.

Addition in , n = , its elements listed , each sum colored by the log of its additive order, 0 white. Selecting a cell lights ; shift-click to select several.

The same ring's products are in Multiplication mod n, and the lifts from $\mathbb{Z}/p^{k-1}$ to $\mathbb{Z}/p^k$ in Lifts mod pᵏ.