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The Montgomery form of the residue class a with respect to R is aR mod N, that is, it is the representative of the residue class aR. For example, suppose that N = 17 and that R = 100. The Montgomery forms of 3, 5, 7, and 15 are 300 mod 17 = 11, 500 mod 17 = 7, 700 mod 17 = 3, and 1500 mod 17 = 4.
Integers in the same congruence class a ≡ b (mod n) satisfy gcd(a, n) = gcd(b, n); hence one is coprime to n if and only if the other is. Thus the notion of congruence classes modulo n that are coprime to n is well-defined.
Fractions: A representation of a non-integer as a ratio of two integers. These include improper fractions as well as mixed numbers . Continued fraction : An expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of ...
The integers arranged on a number line. An integer is the number zero , a positive natural number (1, 2, 3, . . .), or the negation of a positive natural number (−1, −2, −3, . . .). [1] The negations or additive inverses of the positive natural numbers are referred to as negative integers. [2]
The equivalence class modulo m of an integer a is the set of all integers of the form a + k m, where k is any integer. It is called the congruence class or residue class of a modulo m, and may be denoted as (a mod m), or as a or [a] when the modulus m is known from the context.
In its most general formulation, there is a multiset of integers and a target-sum , and the question is to decide whether any subset of the integers sum to precisely . [1] The problem is known to be NP-complete. Moreover, some restricted variants of it are NP-complete too, for example: [1]
Given an integer a and a non-zero integer d, it can be shown that there exist unique integers q and r, such that a = qd + r and 0 ≤ r < | d |. The number q is called the quotient, while r is called the remainder. (For a proof of this result, see Euclidean division. For algorithms describing how to calculate the remainder, see division algorithm.)
The first few prime numbers are 2, 3, 5, 7, and 11. There is no such simple formula as for odd and even numbers to generate the prime numbers. The primes have been widely studied for more than 2000 years and have led to many questions, only some of which have been answered. The study of these questions belongs to number theory.
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