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Is it true that for every $c$ there is a finite set of congruences

$a_i(mod\,\,n_i) , c = n_1<n_2<n_3<...........<n_k \,\,\, (1)\\ $

So that every integer satisfies at least one of the congruence (1)

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You are referring to Covering Systems of congruences. The link, and the name, will let you explore what is a quite large literature. You may also want to look at this survey by Carl Pomerance. If the $n_i$ are strictly increasing, they cannot be chosen arbitrarily.

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  • $\begingroup$ You are welcome. As you can see, it would have been hopeless to summarize! $\endgroup$ Aug 20 '13 at 3:40

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