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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.
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 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 group (/) is cyclic if and only if n is 1, 2, 4, p k or 2p k, where p is an odd prime and k > 0. For all other values of n the group is not cyclic. [1] [2] [3] This was first proved by Gauss. [4] This means that for these n:
When R is a power of a small positive integer b, N′ can be computed by Hensel's lemma: The inverse of N modulo b is computed by a naïve algorithm (for instance, if b = 2 then the inverse is 1), and Hensel's lemma is used repeatedly to find the inverse modulo higher and higher powers of b, stopping when the inverse modulo R is known; N′ is ...
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.
These numerical weights are called probability amplitudes, and this relationship used to calculate probabilities from given pure quantum states (such as wave functions) is called the Born rule. Clearly, the sum of the probabilities, which equals the sum of the absolute squares of the probability amplitudes, must equal 1.
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.