@Deadcode Let A=p^n. Suppose B'>B and C=A (mod B'-1). Since B'>B, it must have p as a factor, so A and B'-1 are coprime. Now we can divide by A: B=1 (mod B'-1). But B'>B, so B=1.
I feel like all of these questions could be easily answered with the use of a couple of carefully chosen Lemmas. But I just do whatever I first think of when I see the problem
@H.PWiz This new second shortened form of division has let me golf 9 bytes off "shift right by half a bit"... and that may just be the beginning. Haven't yet looked through the whole thing for opportunities.