[tex]127\equiv7\mod n[/tex] means there is some positive integer [tex]N[/tex] such that [tex]127=Nn+7[/tex]. Equivalently, there's some [tex]N[/tex] such that [tex]Nn=120[/tex].
This would mean any such number that satisfies the modular congruence must be a positive divisor of 120. There 16 of them:
[tex]\{1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120\}[/tex]
But the congruence doesn't hold for all of these simply because [tex]127\equiv7\mod n[/tex] can't be true for any [tex]n<8[/tex]. This shrinks the pool of candidates to the set
[tex]\{8,10,12,15,20,24,30,40,60,120\}[/tex]
so there are 10 possible solutions for [tex]n[/tex].
