Cayley Table

Each cell gives the row value plus or times the column value, modulo n. Shade encodes the result; outlined cells belong to the identity row or column. Choose a cell to inspect its calculation, identity, inverse or zero divisors. Use arrow keys within the table; Home/End move along a row.

Addition of all residues forms an abelian group with identity 0. Multiplication of all residues forms a commutative monoid with identity 1; 0 has no inverse. Only residues coprime to n have multiplicative inverses. The units form a group; for composite n, nonzero zero divisors occur.