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How to show that these sets are nonempty (here $\mid $ means "divides")?

Here N is an arbitrary large integer and q is some fixed integer.

$R = \lbrace k \in {\mathbb N}:(kN\mid k!) \wedge ((k - 1)N\mid k!) \wedge \cdots \wedge (N\mid k!) \wedge (k > Nq)\rbrace$

$S = \lbrace k \in {\mathbb N}:({(2k - 1)^2}N\mid k!) \wedge ({(2k - 3)^2}N\mid k!) \wedge \ldots \wedge (N\mid k!) \wedge (k > Nq)\rbrace$

$T = \lbrace k \in {\mathbb N}:({k^5}N\mid k!) \wedge ({(k - 1)^5}N\mid k!) \wedge \ldots \wedge (N\mid k!) \wedge (k > Nq)\rbrace$

They exist by the axiom schema of separation, but how do I determine which $k$ to choose so that it satisfies all the properties? Is there a general approach?

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    @Thomas Andrews Never mind. Thanks for help.2012-07-24

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For example, for $R$, you want $k!/(k-j)$ to be a multiple of $N$ for each $j$ from $0$ to $k-1$. That will certainly be true if $k \ge 2N$.

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    Could you explain your reasoning in more detail? How would I apply the same reasoning to $S$ and $T$? Also, I think you mean k>2N because $q$ is an arbitrary integer.2012-07-23