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The equivalence class modulo m of an integer a is the set of all integers of the form a + k m, where k is any integer. It is called the congruence class or residue class of a modulo m, and may be denoted as (a mod m), or as a or [a] when the modulus m is known from the context.
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.
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.
X ≡ 6 (mod 11) has common solutions since 5,7 and 11 are pairwise coprime. A solution is given by X = t 1 (7 × 11) × 4 + t 2 (5 × 11) × 4 + t 3 (5 × 7) × 6. where t 1 = 3 is the modular multiplicative inverse of 7 × 11 (mod 5), t 2 = 6 is the modular multiplicative inverse of 5 × 11 (mod 7) and t 3 = 6 is the modular multiplicative ...
The pattern shown by 8 and 16 holds [6] for higher powers 2 k, k > 2: {,}, is the 2-torsion subgroup, so (/) cannot be cyclic, and the powers of 3 are a cyclic subgroup of order 2 k − 2, so: ( Z / 2 k Z ) × ≅ C 2 × C 2 k − 2 . {\displaystyle (\mathbb {Z} /2^{k}\mathbb {Z} )^{\times }\cong \mathrm {C} _{2}\times \mathrm {C} _{2^{k-2}}.}
In this example, b is 77 digits in length and e is 2 digits in length, but the value b e is 1,304 decimal digits in length. Such calculations are possible on modern computers, but the sheer magnitude of such numbers causes the speed of calculations to slow considerably.
The ray class group modulo m is the quotient C m = I m / i(K m,1). [14] [15] A coset of i(K m,1) is called a ray class modulo m. Erich Hecke's original definition of Hecke characters may be interpreted in terms of characters of the ray class group with respect to some modulus m. [16]
Bulk modulus, a measure of compression resistance; Elastic modulus, a measure of stiffness; Shear modulus, a measure of elastic stiffness; Young's modulus, a specific elastic modulus; Modulo operation (a % b, mod(a, b), etc.), in both math and programming languages; results in remainder of a division; Casting modulus used in Chvorinov's rule.