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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 ...
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.
The multiplicative inverse x ≡ a −1 (mod m) may be efficiently computed by solving Bézout's equation a x + m y = 1 for x, y, by using the Extended Euclidean algorithm. In particular, if p is a prime number, then a is coprime with p for every a such that 0 < a < p ; thus a multiplicative inverse exists for all a that is not congruent to ...
As of 20 August 2020, a video containing the song, misspelt as "Johny" and uploaded to YouTube by Loo Loo Kids in 2016, [1] has more than 6.9 billion views as of January 2024, making it the third-most-viewed video on the site, as well as the most-viewed nursery rhyme video and one of the top 10 most-disliked YouTube videos.
Since taking the square root is the same as raising to the power 1 / 2 , the following is also an algebraic expression: 1 − x 2 1 + x 2 {\displaystyle {\sqrt {\frac {1-x^{2}}{1+x^{2}}}}} An algebraic equation is an equation involving polynomials , for which algebraic expressions may be solutions .
One particular solution is x = 0, y = 0, z = 0. Two other solutions are x = 3, y = 6, z = 1, and x = 8, y = 9, z = 2. There is a unique plane in three-dimensional space which passes through the three points with these coordinates, and this plane is the set of all points whose coordinates are solutions of the equation.
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 ...
Legendre symbol: If p is an odd prime number and a is an integer, the value of () is 1 if a is a quadratic residue modulo p; it is –1 if a is a quadratic non-residue modulo p; it is 0 if p divides a.