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In mathematics, the result of the modulo operation is an equivalence class, and any member of the class may be chosen as representative; however, the usual representative is the least positive residue, the smallest non-negative integer that belongs to that class (i.e., the remainder of the Euclidean division ). [2] However, other conventions are possible. Computers and calculators have various ...
Time-keeping on this clock uses arithmetic modulo 12. Adding 4 hours to 9 o'clock gives 1 o'clock, since 13 is congruent to 1 modulo 12. In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed ...
Finally, given a, the multiplicative inverse of a modulo n is an integer x satisfying ax ≡ 1 (mod n) . It exists precisely when a is coprime to n, because in that case gcd (a, n) = 1 and by Bézout's lemma there are integers x and y satisfying ax + ny = 1. Notice that the equation ax + ny = 1 implies that x is coprime to n, so the multiplicative inverse belongs to the group.
The concept of multiplicative order is a special case of the order of group elements. 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 ...
The Typhon Medium Range Capability (MRC) is a US Army transporter erector launcher for Standard SM-6 and Tomahawk missiles.. Mk 70 Mod 1 Payload Delivery System. The Lockheed Martin Mk 70 Mod 1 is a containerised system containing four strike-length cells from the Mk 41 vertical launch system mounted in the footprint of a 40-foot ISO container.
Modular multiplicative inverse. In mathematics, particularly in the area of arithmetic, a modular multiplicative inverse of an integer a is an integer x such that the product ax is congruent to 1 with respect to the modulus m. [1] In the standard notation of modular arithmetic this congruence is written as.
Here we see that the period of 3 k modulo 7 is 6. The remainders in the period, which are 3, 2, 6, 4, 5, 1, form a rearrangement of all nonzero remainders modulo 7, implying that 3 is indeed a primitive root modulo 7. This derives from the fact that a sequence ( gk modulo n) always repeats after some value of k, since modulo n produces a finite number of values. If g is a primitive root modulo ...
Gauss proved [12] that if p ≡ 1 (mod 4) then the nonzero residue classes (mod p) can be divided into four sets, each containing ( p −1)/4 numbers. Let e be a quadratic nonresidue. The first set is the quartic residues; the second one is e times the numbers in the first set, the third is e2 times the numbers in the first set, and the fourth one is e3 times the numbers in the first set ...