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If p is an odd prime and p − 1 = 2 s d with s > 0 and d odd > 0, then for every a coprime to p, either a d ≡ 1 (mod p) or there exists r such that 0 ≤ r < s and a 2 r d ≡ −1 (mod p). This result may be deduced from Fermat's little theorem by the fact that, if p is an odd prime, then the integers modulo p form a finite field , in which ...
Let k be the order of a, that is, k is the smallest positive integer such that a k ≡ 1 (mod p). Then the numbers 1, a, a 2, ..., a k −1 reduced modulo p form a subgroup of G whose order is k and therefore, by Lagrange's theorem, k divides the order of G, which is p − 1. So p − 1 = km for some positive integer m and then
If p is a prime number which is not a divisor of b, then ab p−1 mod p = a mod p, due to Fermat's little theorem. Inverse: [(−a mod n) + (a mod n)] mod n = 0. b −1 mod n denotes the modular multiplicative inverse, which is defined if and only if b and n are relatively prime, which is the case when the left hand side is defined: [(b −1 ...
AP exams (with few exceptions [1]) have a multiple-choice section and a free-response section. AP Studio Art requires students to submit a portfolio for review. AP Computer Science Principles requires students to complete the Create task, which is part of the AP grade for the class. AP exams were taken by subject in 2013.
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.
Assessment for AP Computer Science Principles is divided into two parts: a Create Performance Task due during the course, as well as an AP exam. [2] AP Computer Science Principles examines a variety of computing topics on a largely conceptual level, and teaches procedural programming. In the Create "Through-Course Assessment", students must ...
Advanced Placement (AP) Environmental Science (also known as APES, AP Enviro, AP Environmental, AP Environment, or AP EnviroSci) is a course and exam offered by the American College Board as part of the Advanced Placement Program to high school students interested in the environmental and natural sciences. AP Environmental Science was first ...