## Quick Reference

(modulo *n*)

For each positive integer *n*, the relation of congruence between integers is defined as follows: *a* is congruent to *b* modulo *n* if *a*−*b* is a multiple of *n*. This is written *a* ≡ *b* (mod *n*). The integer *n* is the modulus of the congruence. Then *a* ≡ *b* (mod *n*) if and only if *a* and *b* have the same remainder upon division by *n*. For example, 19 is congruent to 7 modulo 3. The following properties hold, if *a* ≡ *b* (mod *n*) and *c* ≡ *d* (mod *n*):*a*+*c* ≡ *b*+*d* (mod *n*),*a*−*c* ≡ *b*−*d* (mod *n*),*ac* ≡ *bd* (mod *n*).It can be shown that congruence modulo *n* is an equivalence relation and so defines a partition of the set of integers, where two integers are in the same class if and only if they are congruent modulo *n*. These classes are the residue (or congruence) classes modulo *n*.

*a*+*c* ≡ *b*+*d* (mod *n*),

*a*−*c* ≡ *b*−*d* (mod *n*),

*ac* ≡ *bd* (mod *n*).

*Subjects:*
Mathematics.