The algorithm I used is commented here.

facteurs = First[Transpose[
                    Drop[FactorList[X^24-1],2]]] 
                    
    (* List of irreducible factors, minus (X-1).
    The Φn are created from the procedure FactorList 
    *)
            
    nbreFacteurs = Length[facteurs]

One wishes to partition this list in two subsets, the factors of and . A cool way of doing this is to iterate from 0000…01 to 111…111 in binary digits, the number of digits being the length of the list: from 1 to (half is actually enough, by symmetry).

IntegerDigits[37, 2, nbreFacteurs]

The 1's will pinpoint the position of the factors of in the list, the 0's pinpoint 's.

First /@ Position[IntegerDigits[37, 2, nbreFacteurs], 1]

Map[facteurs[[#]]&, %]

Now we have the first term :

Apply[Times,%]//Expand

(X^24-1)/% // Cancel

Here , hence the result is rejected.