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