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and −2 is the least absolute remainder. In the division of 42 by 5, we have: 42 = 8 × 5 + 2, and since 2 < 5/2, 2 is both the least positive remainder and the least absolute remainder. 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.
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]
Given real numbers x and y, integers m and n and the set of integers, floor and ceiling may be defined by the equations ⌊ ⌋ = {}, ⌈ ⌉ = {}. Since there is exactly one integer in a half-open interval of length one, for any real number x, there are unique integers m and n satisfying the equation
A cell on a different sheet of the same spreadsheet is usually addressed as: =SHEET2!A1 (that is; the first cell in sheet 2 of the same spreadsheet). Some spreadsheet implementations in Excel allow cell references to another spreadsheet (not the currently open and active file) on the same computer or a local network.
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.
A residue numeral system (RNS) is a numeral system representing integers by their values modulo several pairwise coprime integers called the moduli. This representation is allowed by the Chinese remainder theorem, which asserts that, if M is the product of the moduli, there is, in an interval of length M, exactly one integer having any given set of modular values.
If the remainder is 2, swap 1 and 3 in odd list and move 5 to the end (3, 1, 7, 5). If the remainder is 3, move 2 to the end of even list and 1,3 to the end of odd list (4, 6, 8, 2 – 5, 7, 9, 1, 3). Append odd list to the even list and place queens in the rows given by these numbers, from left to right (a2, b4, c6, d8, e3, f1, g7, h5).
The result of this procedure is a number which is smaller than the original whenever the original has more than one digit, leaves the same remainder as the original after division by nine, and may be obtained from the original by subtracting a multiple of 9 from it. The name of the procedure derives from this latter property.