Quotienting is a generalisation of division!

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tgt
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Quotients in abstract algebra is a generalisation of division. While the latter works only for single elements, the former is for dividing sets. Amazing stuff!
 

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mathwonk
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linear combinations are a generalization of multiples, so asking when an element is a linear combination of other elements is a generalization of when it is a multiple of another element, yes.

(a quotient object of course is one where an object is considered zero iff it is a linear combination of given objects, at least in a commutative setting.)
 

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