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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.
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.
Animation showing the use of synthetic division to find the quotient of + + + by .Note that there is no term in , so the fourth column from the right contains a zero.. In algebra, synthetic division is a method for manually performing Euclidean division of polynomials, with less writing and fewer calculations than long division.
See also: List of functional analysis topics, List of wavelet-related transforms; Inverse distance weighting; Radial basis function (RBF) — a function of the form ƒ(x) = φ(|x−x 0 |) Polyharmonic spline — a commonly used radial basis function; Thin plate spline — a specific polyharmonic spline: r 2 log r; Hierarchical RBF
Trial division is the most laborious but easiest to understand of the integer factorization algorithms. The essential idea behind trial division tests to see if an integer n, the integer to be factored, can be divided by each number in turn that is less than or equal to the square root of n.
The notation / is used because this ring is the quotient ring of by the ideal, the set formed by all k m with . Considered as a group under addition, Z / m Z {\displaystyle \mathbb {Z} /m\mathbb {Z} } is a cyclic group , and all cyclic groups are isomorphic with Z / m Z {\displaystyle \mathbb {Z} /m\mathbb {Z} } for some m .
The number q is called the quotient, while r is called the remainder. (For a proof of this result, see Euclidean division. For algorithms describing how to calculate the remainder, see Division algorithm.) The remainder, as defined above, is called the least positive remainder or simply the remainder. [2]
Its existence is based on the following theorem: Given two univariate polynomials a and b ≠ 0 defined over a field, there exist two polynomials q (the quotient) and r (the remainder) which satisfy = + and < (), where "deg(...)" denotes the degree and the degree of the zero polynomial is defined as being negative.