How to show that these sets are nonempty

[glebovg]

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

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

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

${S_{k,q}} = \{ k \in {\mathbb N}:({(2k - 1)^2}N\mid k!) \wedge ({(2k - 3)^2}N\mid k!) \wedge \ldots \wedge (N\mid k!) \wedge (k > Nq)\}$

${T_{k,q}} = \{ k \in {\mathbb N}:({k^5}N\mid k!) \wedge ({(k - 1)^5}N\mid k!) \wedge \ldots \wedge (N\mid k!) \wedge (k > Nq)\}$

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?

 

[lippyka]

One approach is to note that for each set, there is a lower bound on the value of k that will guarantee that the set is nonempty. For R_{k,q}, since (kN\mid k!), we only need to consider k values greater than Nq. Similarly, for S_{k,q} and T_{k,q}, we need to consider k values greater than the square root of Nq and the fifth root of Nq respectively. This ensures that all of the sets are nonempty.