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The multiplicative order of a number a modulo n is the order of a in the multiplicative group whose elements are the residues modulo n of the numbers coprime to n, and whose group operation is multiplication modulo n. This is the group of units of the ring Zn; it has φ (n) elements, φ being Euler's totient function, and is denoted as U (n) or ...
Given two positive numbers a and n, a modulo n (often abbreviated as a mod n) is the remainder of the Euclidean division of a by n, where a is the dividend and n is the divisor. [ 1 ] For example, the expression "5 mod 2" evaluates to 1, because 5 divided by 2 has a quotient of 2 and a remainder of 1, while "9 mod 3" would evaluate to 0 ...
Then one can proceed by adding 20 = 5 × 4 at each step, and computing only the remainders by 3. This gives 4 mod 4 → 0. Continue 4 + 5 = 9 mod 4 →1. Continue 9 + 5 = 14 mod 4 → 2. Continue 14 + 5 = 19 mod 4 → 3. OK, continue by considering remainders modulo 3 and adding 5 × 4 = 20 each time 19 mod 3 → 1. Continue 19 + 20 = 39 mod 3 ...
The W87-1 had a planned first production unit date of July 1997, [5] but Midgetman and W87-1 were canceled in January 1992. In 2019, the W87 mod 1 was selected to replace the W78 warhead deployed on all Minuteman III missiles not currently carrying the W87 mod 0.
The notion of covering system was introduced by Paul Erdős in the early 1930s. The following are examples of covering systems: A covering system is called disjoint (or exact) if no two members overlap. A covering system is called distinct (or incongruent) if all the moduli are different (and bigger than 1). Hough and Nielsen (2019) [1] proved ...
Let p be an odd prime. The quadratic excess E (p) is the number of quadratic residues on the range (0, p /2) minus the number in the range (p /2, p) (sequence A178153 in the OEIS). For p congruent to 1 mod 4, the excess is zero, since −1 is a quadratic residue and the residues are symmetric under r ↔ p − r.
The number of nonresidues found will be even when m ≡ 0, 1 (mod 4), and it will be odd if m ≡ 2, 3 (mod 4). Gauss's fourth proof consists of proving this theorem (by comparing two formulas for the value of Gauss sums) and then restricting it to two primes.
Then () = means that the order of the group is 8 (i.e., there are 8 numbers less than 20 and coprime to it); () = means the order of each element divides 4, that is, the fourth power of any number coprime to 20 is congruent to 1 (mod 20). The set {3,19} generates the group, which means that every element of (/) is of the form 3 a × 19 b (where ...