diff --git a/doc/source/whatsnew/v2.1.0.rst b/doc/source/whatsnew/v2.1.0.rst
index 1eab7b2fc411f..ed85b952e3bcd 100644
--- a/doc/source/whatsnew/v2.1.0.rst
+++ b/doc/source/whatsnew/v2.1.0.rst
@@ -124,6 +124,7 @@ Datetimelike
 Timedelta
 ^^^^^^^^^
 - Bug in :meth:`Timedelta.round` with values close to the implementation bounds returning incorrect results instead of raising ``OutOfBoundsTimedelta`` (:issue:`51494`)
+- Bug in :class:`TimedeltaIndex` division or multiplication leading to ``.freq`` of "0 Days" instead of ``None`` (:issue:`51575`)
 -
 
 Timezones
diff --git a/pandas/core/arrays/timedeltas.py b/pandas/core/arrays/timedeltas.py
index e7a0ddba1fccc..d38145295a4db 100644
--- a/pandas/core/arrays/timedeltas.py
+++ b/pandas/core/arrays/timedeltas.py
@@ -467,6 +467,9 @@ def __mul__(self, other) -> TimedeltaArray:
             freq = None
             if self.freq is not None and not isna(other):
                 freq = self.freq * other
+                if freq.n == 0:
+                    # GH#51575 Better to have no freq than an incorrect one
+                    freq = None
             return type(self)._simple_new(result, dtype=result.dtype, freq=freq)
 
         if not hasattr(other, "dtype"):
@@ -526,17 +529,10 @@ def _scalar_divlike_op(self, other, op):
                 # Note: freq gets division, not floor-division, even if op
                 #  is floordiv.
                 freq = self.freq / other
-
-                # TODO: 2022-12-24 test_ufunc_coercions, test_tdi_ops_attributes
-                #  get here for truediv, no tests for floordiv
-
-                if op is operator.floordiv:
-                    if freq.nanos == 0 and self.freq.nanos != 0:
-                        # e.g. if self.freq is Nano(1) then dividing by 2
-                        #  rounds down to zero
-                        # TODO: 2022-12-24 should implement the same check
-                        #  for truediv case
-                        freq = None
+                if freq.nanos == 0 and self.freq.nanos != 0:
+                    # e.g. if self.freq is Nano(1) then dividing by 2
+                    #  rounds down to zero
+                    freq = None
 
             return type(self)._simple_new(result, dtype=result.dtype, freq=freq)
 
diff --git a/pandas/tests/indexes/timedeltas/test_timedelta.py b/pandas/tests/indexes/timedeltas/test_timedelta.py
index 48dd1fd02d228..1170fd6a053d9 100644
--- a/pandas/tests/indexes/timedeltas/test_timedelta.py
+++ b/pandas/tests/indexes/timedeltas/test_timedelta.py
@@ -157,3 +157,21 @@ def test_freq_conversion(self, index_or_series):
         assert expected.dtype == "m8[s]"
         result = td.astype("timedelta64[s]")
         tm.assert_equal(result, expected)
+
+    def test_arithmetic_zero_freq(self):
+        # GH#51575 don't get a .freq with freq.n = 0
+        tdi = timedelta_range(0, periods=100, freq="ns")
+        result = tdi / 2
+        assert result.freq is None
+        expected = tdi[:50].repeat(2)
+        tm.assert_index_equal(result, expected)
+
+        result2 = tdi // 2
+        assert result2.freq is None
+        expected2 = expected
+        tm.assert_index_equal(result2, expected2)
+
+        result3 = tdi * 0
+        assert result3.freq is None
+        expected3 = tdi[:1].repeat(100)
+        tm.assert_index_equal(result3, expected3)