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It has two definitions: either the integer part of a division (in the case of Euclidean division) [2] or a fraction or ratio (in the case of a general division). For example, when dividing 20 (the dividend ) by 3 (the divisor ), the quotient is 6 (with a remainder of 2) in the first sense and 6 + 2 3 = 6.66... {\displaystyle 6+{\tfrac {2}{3}}=6 ...
Eisenstein discovered that the Fermat quotient with base 2 could be expressed in terms of the sum of the reciprocals modulo p of the numbers lying in the first half of the range {1, ..., p − 1}: − 2 q p ( 2 ) ≡ ∑ k = 1 p − 1 2 1 k ( mod p ) . {\displaystyle -2q_{p}(2)\equiv \sum _{k=1}^{\frac {p-1}{2}}{\frac {1}{k}}{\pmod {p}}.}
For example, the expression "5 mod 2" evaluates to 1, because 5 divided by 2 has a quotient of 2 and a remainder of 1, while "9 mod 3" would evaluate to 0, because 9 divided by 3 has a quotient of 3 and a remainder of 0. Although typically performed with a and n both being integers, many computing systems now allow other types of numeric operands.
This is denoted as 20 / 5 = 4, or 20 / 5 = 4. [2] In the example, 20 is the dividend, 5 is the divisor, and 4 is the quotient. Unlike the other basic operations, when dividing natural numbers there is sometimes a remainder that will not go evenly into the dividend; for example, 10 / 3 leaves a remainder of 1, as 10 is not a multiple of 3.
Using Euclidean division, 9 divided by 4 is 2 with remainder 1. In other words, each person receives 2 slices of pie, and there is 1 slice left over. This can be confirmed using multiplication, the inverse of division: if each of the 4 people received 2 slices, then 4 × 2 = 8 slices were given out in total.
12 / 135 \ 11,25 12 15 12 30 24 60 60 0 In Finland , the Italian method detailed above was replaced by the Anglo-American one in the 1970s. In the early 2000s, however, some textbooks have adopted the German method as it retains the order between the divisor and the dividend.
When dividing 12 by 3 one obtains the result 4 because one can regroup 12 objects into 4 subcollections of 3 objects. The quotient group is the same idea, although one ends up with a group for a final answer instead of a number because groups have more structure than an arbitrary collection of objects: in the quotient G / N {\displaystyle G ...
The difference quotient is sometimes also called the Newton quotient [10] [12] [13] [14] (after Isaac Newton) or Fermat's difference quotient (after Pierre de Fermat). [ 15 ] Overview