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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.
So we can use the remainder from casting out ninety nines to get the remainder of division by eleven. This is called casting out elevens. The same result can also be calculated directly by alternately adding and subtracting the digits that make up . Eleven divides if and only if eleven divides that sum. [7]
The division with remainder or Euclidean division of two natural numbers provides an integer quotient, which is the number of times the second number is completely contained in the first number, and a remainder, which is the part of the first number that remains, when in the course of computing the quotient, no further full chunk of the size of ...
In arithmetic, short division is a division algorithm which breaks down a division problem into a series of easier steps. It is an abbreviated form of long division — whereby the products are omitted and the partial remainders are notated as superscripts.
Chinese remainder theorem; Fermat's little theorem. Proofs of Fermat's little theorem; Fermat quotient; Euler's totient function. Noncototient; Nontotient; Euler's theorem; Wilson's theorem; Primitive root modulo n. Multiplicative order; Discrete logarithm; Quadratic residue. Euler's criterion; Legendre symbol; Gauss's lemma (number theory ...
Thus, the function may be more "cheaply" evaluated using synthetic division and the polynomial remainder theorem. The factor theorem is another application of the remainder theorem: if the remainder is zero, then the linear divisor is a factor. Repeated application of the factor theorem may be used to factorize the polynomial.
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