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For cancellation of common terms, we have the following rules: If a + k ≡ b + k (mod m), where k is any integer, then a ≡ b (mod m). If k a ≡ k b (mod m) and k is coprime with m, then a ≡ b (mod m). If k a ≡ k b (mod k m) and k ≠ 0, then a ≡ b (mod m). The last rule can be used to move modular arithmetic into division.
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.
The set of integers modulo 2 has just two elements; the addition operation it inherits is known in Boolean logic as the "exclusive or" function. A similar "wrap around" operation arises in geometry , where the sum of two angle measures is often taken to be their sum as real numbers modulo 2π.
The set of (congruence classes of) integers modulo n with the operations of addition and multiplication is a ring. It is denoted Z / n Z {\displaystyle \mathbb {Z} /n\mathbb {Z} } or Z / ( n ) {\displaystyle \mathbb {Z} /(n)} (the notation refers to taking the quotient of integers modulo the ideal n Z {\displaystyle n\mathbb {Z} } or ( n ...
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.
Note that is equivalent to zero in the above equation because addition of coefficients is performed modulo 2: = + = (+) = (). Polynomial addition modulo 2 is the same as bitwise XOR. Since XOR is the inverse of itself, polynominal subtraction modulo 2 is the same as bitwise XOR too.
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 ...
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.