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How is it true that:

If $a_1, a_2,\ldots,a_n$ are pairwise relatively prime positive integers,

then $M_i = \dfrac{(a_1a_2\cdots a_n)}{a_i} $ is relatively prime to $a_i$ ?

This is supposed to be an obvious step in the proof of Chinese remainder theorem, but to me it is not obvious.

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    Note that $M_i$ is the product of all the $a_j$ **except** for $a_i$. Now use the fact that the product of numbers relatively prime to $a$ is relatively prime to $a$.2012-02-07
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    Note: "primal" is actually a technical term in ring theory, distinct from "prime". "Relative primality" can mean something different in general context.2012-02-07
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    @Arturo If you refer to Cohn's notion of [primal,](http://math.stackexchange.com/a/99313/23500) then I don't think there is any danger of confusion, since "relatively primal" is not in use (what could it mean?)2012-02-07
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    @MathGems: In a commutative ring, an ideal $Q$ is primal if all elements that are not relatively prime to $Q$ form an ideal. An element $a$ is relatively prime to $Q$ if $ab\in Q$ implies $b\in Q$. Since two elements are "relatively prime" if there is no prime ideal that contains both, they would be "relatively primal" if there is no primal ideal that contains both.2012-02-07
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    @Arturo That meaning of "relatively primal" does not appear to be in use, so, again, there does not appear to be any danger of confusion.2012-02-07
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    @MathGems: "Relative primality" is not in standard use either, though I agree it is more grammatical. However, my ring theory colleagues objected strenously when I noted, in a thesis, that grammatically one should use "primality" in some instances when refering to issues relating to prime (E.g., prove "relative primality" instead of "relative primeness", "prove primality" instead of "prove prime"); the objection was precisely that since "primal" has a specific meaning, and the terms such as "relative primality" are not standard, it is bound to cause confusion.2012-02-07
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    @Arturo The OP's "relative primality" is widely used, as a google search quickly confirms, e.g. it is used by your advisor G. Bergman in "Right orderable groups that are not locally indicable", by K. Ribet in "Hodge classes on certain types of abelian varieties", by K. Conrad in "Selmer's example", by H. Davenport in "Notes on a result of Chalk", etc.2012-02-07
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    @MathGems: Perhaps my ring theory colleagues misinformed me, then.2012-02-07

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