nat division by timedelta

doc/source/whatsnew/v0.21.0.txt

@@ -830,6 +830,7 @@ Other API Changes
 - Restricted DateOffset keyword arguments.  Previously, ``DateOffset`` subclasses allowed arbitrary keyword arguments which could lead to unexpected behavior.  Now, only valid arguments will be accepted. (:issue:`17176`).
 - Pandas no longer registers matplotlib converters on import. The converters
   will be registered and used when the first plot is draw (:issue:`17710`)
+- ``NaT`` division with :class:`datetime.timedelta` will now return ``NaN`` instead of raising (:issue:`17876`)
 
 .. _whatsnew_0210.deprecations:
 

pandas/_libs/tslib.pyx

@@ -1428,15 +1428,15 @@ _nat_scalar_rules[Py_GE] = False
 
 
 cdef _nat_divide_op(self, other):
-    if (isinstance(other, Timedelta) or
+    if (PyDelta_Check(other) or
         is_timedelta64_object(other) or other is NaT):
         return np.nan
     if is_integer_object(other) or is_float_object(other):
         return NaT
     return NotImplemented
 
 cdef _nat_rdivide_op(self, other):
-    if isinstance(other, Timedelta):
+    if PyDelta_Check(other):
         return np.nan
     return NotImplemented
 

pandas/tests/scalar/test_nat.py

@@ -247,6 +247,8 @@ def test_nat_arithmetic():
         assert left + right is NaT
         assert right - left is NaT
         assert left - right is NaT
+        assert np.isnan(left / right)
+        assert np.isnan(right / left)
 
     # GH 11718
     t_utc = Timestamp('2014-01-01', tz='UTC')