John, there seems to be a confusion in your answer.

The current behavior of (p/q) % n is to return Zmod(n)(p/q).lift(). We
want to change it to be something else as proposed in [1], [2] and
[3]. What I want is to preserve a way of doing efficiently the old
behavior...

[1] https://groups.google.com/forum/#!topic/sage-devel/PfMop0nyiL0
[2] https://trac.sagemath.org/ticket/21745
[3] https://trac.sagemath.org/ticket/15260

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