# “the product of the factors” versus “the factors of the product”

Could somebody please compare and contrast the meanings of the two phrases:

"the product of the factors"

and

"the factors of the product."

In terms of expressing possession. Thank you.

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In what context. please give an example. Multipliation and division in a commutative situation, non-commutative? – plusepsilon.de Apr 20 '12 at 14:09
@late_learner Algebra-precalculus arithmetic teaching is the context in a commutative situation. – Sᴋᴜʟʟ ᴘᴇᴛʀᴏʟ Apr 20 '12 at 14:17
Neither really expresses possession, though the second comes closer. The factors don't really belong to the product, but they do go into its construction. Similarly, in the first the product doesn't belong to the factors, though it is the result of an operation on them. Brian M. Scott – Sᴋᴜʟʟ ᴘᴇᴛʀᴏʟ May 2 '12 at 21:03
They're symbiotes, and it's a question of whose point of view you adopt. – Sᴋᴜʟʟ ᴘᴇᴛʀᴏʟ May 2 '12 at 21:16

## 3 Answers

The factors go into the product's construction. Also, the product is the result of an operation on the factors.

The same can be said for a sum and its terms:

The terms go into the sum's construction. Also, the sum is the result of an operation on the terms.

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The factors $a$, $b$, and $c$ have product $a\cdot b\cdot c$.

$a$, $b$, and $c$ are the factors of the product $abc$.

$abc$ is the product of the factors $a$, $b$, and $c$.

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I think "the product of the factors" is a value with each factor in the exact expressing possession.And "the factors of the product." is a set of factor values, that means you can express these values in any order.

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