My precedent post has a big flaw, which I noticed after the last posts of Charlie and Harry, so I'm correcting it here.

C=5*(F-32)/9  

For F congruent 5 (mod 9) C will be an integer  (congruent to mod 5).

C/F will be integer when |C| higher or equal than |F| (or if C=0). For F values, this occurs in the interval: (-40, 11). There are six possible values of F mod compliant in that interval: (-40, -31, -22, -13,- 4, 5)

F=-40 C=-40 N=1

F=-31 C=-35 N not integer

F=-22 C=-30 N=not integer

F=-13 C=-25 N=not integer

F=-4   C=-20 N=5

F=5    C=-15 N=-3

and the solution for C=0 (F=32, N=0)

So N,X = (1,-40) (5,-20) (-3,-15) (0,0)

Charlie points out two other solutions admitting that Celsius degrees and N are integers but not necessarily the Farenheit degrees. It could be.

But really the formula of the puzzle is not precise as X° is symbol for both Celsius degrees and Farenheit degrees. All of us have used X° as Celsius, but we could have used X° for Farenheit, and could had given our answers also as (1,-40) (5,-4) (3,-5) (0,0), referring the numbers to N and Farenheit degrees. It would have been also a valid answer. 

So perhaps, as it is, (N, X) integers stands for N, F, C integers, but if X stands only for C, which is in the logic of the puzzle, the set of "simple" F solutions can be enlarged with other two values.