How to show that these sets are nonempty ([itex]\mid [/itex] means "divides")?(adsbygoogle = window.adsbygoogle || []).push({});

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

[itex]{R_{k,q}} = \{ k \in {\mathbb N}:(kN\mid k!) \wedge ((k - 1)N\mid k!) \wedge \cdots \wedge (N\mid k!) \wedge (k > Nq)\}[/itex]

[itex]{S_{k,q}} = \{ 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)\}[/itex]

[itex]{T_{k,q}} = \{ 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)\}[/itex]

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

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# Homework Help: How to show that these sets are nonempty

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