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A pseudo-remainder sequence is the sequence of the (pseudo) remainders r i obtained by replacing the instruction +:= (,) of Euclid's algorithm by +:= (,), where α is an element of Z that divides exactly every coefficient of the numerator.
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]
Shift a one bit to the left, discarding the leftmost bit, and making the new rightmost bit zero. This multiplies the polynomial by x, but we still need to take account of carry which represented the coefficient of x 7. If carry had a value of one, exclusive or a with the hexadecimal number 0x1b (00011011 in binary).
Animation showing an application of the Euclidean algorithm to find the greatest common divisor of 62 and 36, which is 2. A more efficient method is the Euclidean algorithm, a variant in which the difference of the two numbers a and b is replaced by the remainder of the Euclidean division (also called division with remainder) of a by b.
In these examples, the (negative) least absolute remainder is obtained from the least positive remainder by subtracting 5, which is d. This holds in general. When dividing by d, either both remainders are positive and therefore equal, or they have opposite signs. If the positive remainder is r 1, and the negative one is r 2, then r 1 = r 2 + d.
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.
Long division is the standard algorithm used for pen-and-paper division of multi-digit numbers expressed in decimal notation. It shifts gradually from the left to the right end of the dividend, subtracting the largest possible multiple of the divisor (at the digit level) at each stage; the multiples then become the digits of the quotient, and the final difference is then the remainder.
The second difference lies in the necessity of defining how one complex remainder can be "smaller" than another. To do this, a norm function f ( u + vi ) = u 2 + v 2 is defined, which converts every Gaussian integer u + vi into an ordinary integer.