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In arithmetic, a quotient (from Latin: quotiens "how many times", pronounced ˈkwoʊʃənt) is the quantity produced by the division of two numbers. For example, when dividing twenty (the dividend) by three (the divisor), the quotient is six and two thirds. In this sense, a quotient is the ratio of a dividend to its divisor.
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Integer part definition
The quotient is also less commonly defined as the greatest whole number of times a divisor may be subtracted from a dividend without the remainder becoming negative. For example, the divisor three may be subtracted up to six times from the dividend twenty before the remainder becomes negative: 20-3-3-3-3-3-3 ≥ 0, while 20-3-3-3-3-3-3-3 < 0. In this sense, a quotient is the integer part of the ratio of two numbers.
Other usages
Outside of arithmetic, many branches of mathematics have borrowed the word "quotient" to describe structures built by breaking larger structures into pieces. Given a set with an equivalence relation defined on it, a "quotient set" may be created which contains those equivalence classes as elements. A quotient group may be formed by breaking a group into a number of similar cosets, while a quotient space may be formed in a similar process by breaking a vector space into a number of similar linear subspaces.