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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 by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae , published in 1801.
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, called the modulus of the operation.. 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.
In modular arithmetic, the integers coprime (relatively prime) to n from the set {,, …,} of n non-negative integers form a group under multiplication modulo n, called the multiplicative group of integers modulo n.
The congruence relation, modulo m, partitions the set of integers into m congruence classes. Operations of addition and multiplication can be defined on these m objects in the following way: To either add or multiply two congruence classes, first pick a representative (in any way) from each class, then perform the usual operation for integers on the two representatives and finally take the ...
In modular arithmetic, the modular additive inverse of x is the number a such that a + x ≡ 0 (mod n) and always exists. For example, the inverse of 3 modulo 11 is 8, as 3 + 8 ≡ 0 (mod 11). [12] In a Boolean ring, which has elements {,} addition is often defined as the symmetric difference.
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 1 modulo p has exactly two square roots, 1 and −1 modulo p. Note that a d ≡ 1 (mod p) holds trivially for a ≡ 1 (mod p), because the congruence relation is compatible with exponentiation.
The most direct method of calculating a modular exponent is to calculate b e directly, then to take this number modulo m. Consider trying to compute c, given b = 4, e = 13, and m = 497: c ≡ 4 13 (mod 497) One could use a calculator to compute 4 13; this comes out to 67,108,864. Taking this value modulo 497, the answer c is determined to be 445.
In mathematics, modular arithmetic is a system of arithmetic for certain equivalence classes of integers, called congruence classes. Sometimes it is suggestively called 'clock arithmetic', where numbers 'wrap around' after they reach a certain value (the modulus). For example, when the modulus is 12, then any two numbers that leave the same ...