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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.
This implies that the multiplication is associative, commutative, and that the class of 1 is the unique multiplicative identity. Finally, given a , the multiplicative inverse of a modulo n is an integer x satisfying ax ≡ 1 (mod n ) .
Conditions 1, 2, and 3 say that ~ is an equivalence relation. A congruence ~ is determined entirely by the set {a ∈ G | a ~ e} of those elements of G that are congruent to the identity element, and this set is a normal subgroup. Specifically, a ~ b if and only if b −1 * a ~ e.
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.
Modulo is a mathematical jargon that was introduced into mathematics in the book Disquisitiones Arithmeticae by Carl Friedrich Gauss in 1801. [3] Given the integers a, b and n, the expression "a ≡ b (mod n)", pronounced "a is congruent to b modulo n", means that a − b is an integer multiple of n, or equivalently, a and b both share the same remainder when divided by n.
The first known proof is attributed to Axel Thue [2] who used a pigeonhole argument. [3] It can be used to prove Fermat's theorem on sums of two squares by taking m to be a prime p that is congruent to 1 modulo 4 and taking a to satisfy a 2 + 1 ≡ 0 mod p. (Such an "a" is guaranteed for "p" by Wilson's theorem. [4])
Arithmetic topology is an area of mathematics that is a combination of algebraic number theory and topology. It establishes an analogy between number fields and closed, orientable 3-manifolds . Analogies
The output of the integer operation determines a residue class, and the output of the modular operation is determined by computing the residue class's representative. For example, if N = 17 , then the sum of the residue classes 7 and 15 is computed by finding the integer sum 7 + 15 = 22 , then determining 22 mod 17 , the integer between 0 and ...