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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. [1]
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.
If a modulus just less than a power of 2 is used (the Mersenne primes 2 31 − 1 and 2 61 − 1 are popular, as are 2 32 − 5 and 2 64 − 59), reduction modulo m = 2 e − d can be implemented more cheaply than a general double-width division using the identity 2 e ≡ d (mod m).
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 ...
Modular exponentiation is the remainder when an integer b (the base) is raised to the power e (the exponent), and divided by a positive integer m (the modulus); that is, c = b e mod m. From the definition of division, it follows that 0 ≤ c < m .
For computational purposes it is also necessary that division and reduction modulo R are inexpensive, and the modulus is not useful for modular multiplication unless R > N. The Montgomery form of the residue class a with respect to R is aR mod N , that is, it is the representative of the residue class aR .
MicroPython is a software implementation of a programming language largely compatible with Python 3, written in C, that is optimized to run on a microcontroller. [ 2 ] [ 3 ] MicroPython consists of a Python compiler to bytecode and a runtime interpreter of that bytecode.
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.