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Let and be integers with . Then unique integers and with the property that , where .

For any nonzero integers and integers and gcd(). Moreover, gcd() is the smallest positive integer of the form .

if p is prime then p|ab implies p|a and/or p|b.

Every integer greater than 1 is a prime or the unique product of primes.

Let be a set of integers containing . Suppose has the property that whenever some integer , then the integer . Then, contains every integer greater than or equal to .

Let be a set of integers containing . Suppose has the property that whenever every integer less than and greater than or equal to belongs to . Then, contains every integer greater than or equal to

An equivelence relation on a set S is a set R of ordered pairs ot elements of S such that:
( (reflexive)
implies (symmetric)
and imply (transitive)

A partition of a set is a collection of nonempty disjoint subsets of whose unions is .

The equivalence classes of an equivalence relation on a set constitute a partition of . Conversely, for any partition of , there is an equivalence relation on whose equivalence classes are the elements of

A function (or mapping) from a set to a set is a rule that assigns to each element of exactly one element of . The set is called the domain of and is called the range of . If assigns to , then is called the image of under . The subset of comprising all the images of elements of is called the image of A under .