What do Chaya and Hannah both have in common in The Devil's Arithmetic?

In the simplest way in The Devil’s Arithmetic, Chaya and Hannah have their Jewish faith in common (which includes all of the pertinent Jewish traditions).  However, one must realize that Hannah actually “becomes” Chaya when Hannah goes back in time.  Chaya and Hannah are actually the same character for much of the book.  When Hannah “becomes” Chaya (as Hannah opens the door and sees a field instead of the other apartments), Hannah has the same experiences and gains many of the same traits as Chaya, such as compassion and bravery.  For example, Hannah/Chaya shows compassion when she gives her only bread ration to some younger Jewish children in the concentration camp.  Most importantly, though, Hannah/Chaya shows bravery when she decides to sacrifice herself for Rivka, asking Rivka only to “run” and “remember.”  Hannah/Chaya then walks steadfastly into the gas chamber and is killed.  This, of course, is the climax of The Devil’s Arithmetic.