diff --git a/doc/source/whatsnew/v0.24.0.rst b/doc/source/whatsnew/v0.24.0.rst
index 0b2b526dfe9e7..488971c13508a 100644
--- a/doc/source/whatsnew/v0.24.0.rst
+++ b/doc/source/whatsnew/v0.24.0.rst
@@ -1516,6 +1516,7 @@ Groupby/Resample/Rolling
 
 - Bug in :func:`pandas.core.groupby.GroupBy.first` and :func:`pandas.core.groupby.GroupBy.last` with ``as_index=False`` leading to the loss of timezone information (:issue:`15884`)
 - Bug in :meth:`DatetimeIndex.resample` when downsampling across a DST boundary (:issue:`8531`)
+- Bug in date anchoring for :meth:`DatetimeIndex.resample` with offset :class:`Day` when n > 1 (:issue:`24127`)
 - Bug where ``ValueError`` is wrongly raised when calling :func:`~pandas.core.groupby.SeriesGroupBy.count` method of a
   ``SeriesGroupBy`` when the grouping variable only contains NaNs and numpy version < 1.13 (:issue:`21956`).
 - Multiple bugs in :func:`pandas.core.Rolling.min` with ``closed='left'`` and a
diff --git a/pandas/core/resample.py b/pandas/core/resample.py
index dc1f94c479a37..6d80d747f21b3 100644
--- a/pandas/core/resample.py
+++ b/pandas/core/resample.py
@@ -1587,8 +1587,8 @@ def _get_range_edges(first, last, offset, closed='left', base=0):
         is_day = isinstance(offset, Day)
         day_nanos = delta_to_nanoseconds(timedelta(1))
 
-        # #1165
-        if (is_day and day_nanos % offset.nanos == 0) or not is_day:
+        # #1165 and #24127
+        if (is_day and not offset.nanos % day_nanos) or not is_day:
             return _adjust_dates_anchored(first, last, offset,
                                           closed=closed, base=base)
 
diff --git a/pandas/tests/groupby/test_timegrouper.py b/pandas/tests/groupby/test_timegrouper.py
index 183ccfb5182a2..cb7b419710837 100644
--- a/pandas/tests/groupby/test_timegrouper.py
+++ b/pandas/tests/groupby/test_timegrouper.py
@@ -43,8 +43,8 @@ def test_groupby_with_timegrouper(self):
 
             expected = DataFrame(
                 {'Quantity': 0},
-                index=date_range('20130901 13:00:00',
-                                 '20131205 13:00:00', freq='5D',
+                index=date_range('20130901',
+                                 '20131205', freq='5D',
                                  name='Date', closed='left'))
             expected.iloc[[0, 6, 18], 0] = np.array([24, 6, 9], dtype='int64')
 
diff --git a/pandas/tests/resample/test_datetime_index.py b/pandas/tests/resample/test_datetime_index.py
index b287eb468cd94..69fb92486d523 100644
--- a/pandas/tests/resample/test_datetime_index.py
+++ b/pandas/tests/resample/test_datetime_index.py
@@ -1463,3 +1463,29 @@ def f(data, add_arg):
         result = df.groupby("A").resample("D").agg(f, multiplier)
         expected = df.groupby("A").resample('D').mean().multiply(multiplier)
         assert_frame_equal(result, expected)
+
+    @pytest.mark.parametrize('k', [1, 2, 3])
+    @pytest.mark.parametrize('n1, freq1, n2, freq2', [
+        (30, 'S', 0.5, 'Min'),
+        (60, 'S', 1, 'Min'),
+        (3600, 'S', 1, 'H'),
+        (60, 'Min', 1, 'H'),
+        (21600, 'S', 0.25, 'D'),
+        (86400, 'S', 1, 'D'),
+        (43200, 'S', 0.5, 'D'),
+        (1440, 'Min', 1, 'D'),
+        (12, 'H', 0.5, 'D'),
+        (24, 'H', 1, 'D'),
+    ])
+    def test_resample_equivalent_offsets(self, n1, freq1, n2, freq2, k):
+        # GH 24127
+        n1_ = n1 * k
+        n2_ = n2 * k
+        s = pd.Series(0, index=pd.date_range('19910905 13:00',
+                                             '19911005 07:00',
+                                             freq=freq1))
+        s = s + range(len(s))
+
+        result1 = s.resample(str(n1_) + freq1).mean()
+        result2 = s.resample(str(n2_) + freq2).mean()
+        assert_series_equal(result1, result2)