Every nonempty set of positive integers contains a smallest number.
Let a and b be integers with b > 0. Then there exist unique integers a and r with the property that a = bq +r where 0<=r<=b
Greatest Common Divisor/Relatively Prime Integers
The greatest common divisor of two nonzero integers a and b is the largest of all common divisors of a and b. We denote this integer by gcd(a,b). When gcd(a,b) = 1, we say a and b are relatively prime.
a mod n
Let n be a fixed positive integer. For any integer a, a mod n is the remainder upon dividing a by n.
If a and b are integers and n is a positive integer, we write a = b mod n when n divides a - b.
First Principle of Mathematical Induction
Let S be a set of integers containing a. Suppose S has the property that whenever some integer n >= a belongs to S, then the integer n + 1 also belongs to S. Then, S contains every integer greater than or equal to a.
Second Principle of Mathematical Induction
Let S be a set of integers containing a. Suppose S has the property that n belongs to S whenever every integer less than n and greater than or equal to a belongs to S. Then, S contains every integer greater than or equal to a.
An equivalence relation on a set S is a set R of ordered pairs of elements such that the reflexive, symmetric, and transitive properties hold.
A partition of a set S is a collection of nonempty disjoint subsets of S whose union is S.
A function k (or mapping) from a set A to a set B is a rule that assigns to each element a of A exactly one element b of B. (Definition Shortened)
Composition of Functions
let k: A-->B and j:B-->C. The composition jk is the mapping from A to C defined by (jk)(a)=j(k(a)) for all a in A.
A function k from a set A to B is called one-to-one is k(a1) = k(a2) implies a1=a2.
Function from A onto B
A function k from a set A to a set B is said to be onto B if each element of B is the image of a least one element of A.
Let G be a set. A binary operation on G is a function that assigns each ordered pair of elements of G an element of G.
Let G be a nonempty set together with a binary operation that assigns each ordered pair (a,b) of elements of G an element in G denoted by ab. We say G is a group under this operation if it is associative, has an identity, and inverses. (Shortened)
If a group has the property that ab =ba for every pair of elements a and b, we say the group is Abelian.
Least Common Multiple
The least common multiple of two nonzero integers a and b is the smallest positive integer that is a multiple of both a and b. We will denote this integer by lcm(a,b).
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