How can i make FullSimplify work nice with Mod ?

Questions
,
1

These codes are not given the correct answer

FullSimplify[Mod[b - 1, b], Element[b, Integers] && b > 1]

FullSimplify[Mod[(b - 1)! + 1, b] == 0, Element[b, Primes]]
2

What results do you get when you evaluate those?

What results do you expect?