diff --git a/Lib/_pydecimal.py b/Lib/_pydecimal.py
index 9b8e42a2342536..9fdecf68e89c5b 100644
--- a/Lib/_pydecimal.py
+++ b/Lib/_pydecimal.py
@@ -442,6 +442,95 @@ def IEEEContext(bits, /):
 
 ##### Decimal class #######################################################
 
+# Observation: For all q >= 0 and a >= 1, q < 10**a iff len(str(q)) <= a.
+#
+# The constants below are used to speed-up "q < 10 ** a" checks to avoid
+# computing len(str(q)) as much as possible. Those speed-ups are based on
+# the following claims.
+#
+# See https://github.com/python/cpython/issues/140036 for details.
+
+_LOG_10_BASE_2_LO = float.fromhex('0x1.a934f0979a371p+1')
+assert pow(2, _LOG_10_BASE_2_LO) < 10
+
+_LOG_10_BASE_2_HI = float.fromhex('0x1.a934f0979a372p+1')
+assert pow(2, _LOG_10_BASE_2_HI) > 10
+
+
+def _tento(n):
+    """Compute 10 ** n with 1 base-5 exponentiation and 1 bit-shift."""
+    return (5 ** n) << n
+
+
+def _is_less_than_pow10a_use_str(q, a):
+    """Try to efficiently check len(str(q)) <= a, or equivalently q < 10**a.
+
+    If the comparison cannot be obtained from q.bit_length(),
+    then str(q) is explicitly computed and may raise ValueError.
+
+    Return (len(str(q)) <= a, None) or (len(str(q)) <= a, str(q)).
+    """
+    if q.bit_length() < a * _LOG_10_BASE_2_LO:
+        # Claim: If 0 < z <= log2(10) and q.bit_length() < a*z, then q < 10**a.
+        # Proof: By contradiction, q >= 10**a. By definition,
+        #   log2(q) >= a*log2(10) >= a*z > q.bit_length().
+        # In particular, q > 2**q.bit_length(), which is impossible.
+
+        # assert q < 10 ** a
+        return True, None
+    elif q.bit_length() >= 1 + a * _LOG_10_BASE_2_HI:
+        # Claim: If z > log2(10) and q.bit_length() >= 1 + a*z, then q > 10**a.
+        # Proof: Since q >= 2**(q.bit_length()-1), we have
+        #   q >= 2**(q.bit_length()-1) >= 2**(a*z) > 2**(a*log2(10)) = 10**a.
+
+        # assert q > 10 ** a
+        return False, None
+    # Handle cases that fail due to floating point precision loss
+    # when computing _LOG_10_BASE_2_LO and _LOG_10_BASE_2_HI, or
+    # that cannot be distinguished with (q.bit_length(), a) only.
+    #
+    # For instance, (q1, a) = (95, 2) and (q2, a) = (105, 2) produce
+    # different results but q1.bit_length() == q2.bit_length() == 7.
+    str_q = str(q)  # can raise a ValueError
+    is_valid = len(str_q) <= a
+    return is_valid, str_q
+
+
+def _is_less_than_pow10a(q, a, *, exact=True, ulp_order=20):
+    """Check that len(str(q)) <= a without computing str(q).
+
+    When *exact* is false, computing len(str(q)) is replaced by f(q):
+
+        f(q) = floor(log10(q) + ulp(log10(q)) * ulp_order + 1.0)
+
+    Most of the time, f(q) = len(str(q)) but in some cases, it may
+    happen that f(q) > len(str(q)).
+
+    When *exact* is true, computing len(str(q)) requires one bigint
+    exponentiation that only depends on q.
+    """
+    if q < 10:
+        return a >= 1
+
+    z = _math.log10(q)
+    t = _math.ulp(z) * ulp_order
+
+    if exact:
+        intlo = int(z - t)
+        inthi = int(z + t)
+        diff = inthi - intlo
+        assert diff in (0, 1)
+        if diff == 1:
+            lo = _tento(inthi)  # may be slow
+            if q < lo:
+                inthi -= 1
+                assert q >= (lo // 10)
+        ndigits = inthi + 1
+    else:
+        ndigits = int(z + t + 1.0)
+    return ndigits <= a
+
+
 # Do not subclass Decimal from numbers.Real and do not register it as such
 # (because Decimals are not interoperable with floats).  See the notes in
 # numbers.py for more detail.
@@ -1355,8 +1444,13 @@ def _divide(self, other, context):
             else:
                 op2.int *= 10**(op2.exp - op1.exp)
             q, r = divmod(op1.int, op2.int)
-            if q < 10**context.prec:
-                return (_dec_from_triple(sign, str(q), 0),
+            is_valid, str_q = _is_less_than_pow10a_use_str(q, context.prec)
+            if is_valid:
+                if str_q is None:
+                    str_q = str(q)
+                # assert q < 10 ** context.prec
+                # assert len(str(q)) <= context.prec
+                return (_dec_from_triple(sign, str_q, 0),
                         _dec_from_triple(self._sign, str(r), ideal_exp))
 
         # Here the quotient is too large to be representable
@@ -1511,7 +1605,9 @@ def remainder_near(self, other, context=None):
             r -= op2.int
             q += 1
 
-        if q >= 10**context.prec:
+        if not _is_less_than_pow10a(q, context.prec):
+            # assert q >= 10 ** context.prec
+            # assert len(str(q)) > context.prec
             return context._raise_error(DivisionImpossible)
 
         # result has same sign as self unless r is negative
diff --git a/Lib/test/test_decimal.py b/Lib/test/test_decimal.py
index 08a8f4c3b36bd6..7eeb77dec78e0c 100644
--- a/Lib/test/test_decimal.py
+++ b/Lib/test/test_decimal.py
@@ -24,6 +24,7 @@
 with the corresponding argument.
 """
 
+import contextlib
 import logging
 import math
 import os, sys
@@ -2611,6 +2612,37 @@ def tearDown(self):
         sys.set_int_max_str_digits(self._previous_int_limit)
         super().tearDown()
 
+    def test_helper__is_less_than_pow10a_use_str_slow_path(self):
+        # Test the "slow" path of _is_less_than_pow10a_use_str().
+        a, b = 2, 7
+
+        # Choose q1, q2 such that len(str(q1)) <= a < len(str(q2))
+        # and q1.bit_length() == q2.bit_length() == b to check that
+        # we cover the "slow" path correctly even for small values.
+        q1, q2 = 95, 105
+        b1, b2 = q1.bit_length(), q2.bit_length()
+
+        self.assertEqual(b1, b)
+        self.assertEqual(b2, b)
+
+        # ensure that the first "fast" check doesn't hold
+        self.assertGreaterEqual(b, a * self.decimal._LOG_10_BASE_2_LO)
+        # ensure that the second "fast" check doesn't hold
+        self.assertLess(b, 1 + a * self.decimal._LOG_10_BASE_2_HI)
+
+        cond_q1, str_q1 = self.decimal._is_less_than_pow10a_use_str(q1, a)
+        self.assertTrue(cond_q1)
+        self.assertIsNotNone(str_q1)
+
+        cond_q2, str_q2 = self.decimal._is_less_than_pow10a_use_str(q2, a)
+        self.assertFalse(cond_q2)
+        self.assertIsNotNone(str_q2)
+
+    def test_helper__is_less_than_pow10a(self):
+        # TODO(picnixz): find a simple test case with custom ulp_order.
+        pass
+
+
 class PythonAPItests:
 
     def test_abc(self):
@@ -4493,6 +4525,15 @@ def test_decimal_attributes(self):
 
 class Coverage:
 
+    @contextlib.contextmanager
+    def unbound_context(self, prec=None, Emax=None, Emin=None):
+        with self.decimal.localcontext() as c:
+            c.prec = self.decimal.MAX_PREC if prec is None else prec
+            c.Emax = self.decimal.MAX_EMAX if Emax is None else Emax
+            c.Emin = self.decimal.MIN_EMIN if Emin is None else Emin
+            c.traps[self.decimal.Inexact] = 1
+            yield c
+
     def test_adjusted(self):
         Decimal = self.decimal.Decimal
 
@@ -4660,6 +4701,22 @@ def test_divmod(self):
             self.assertTrue(c.flags[InvalidOperation] and
                             c.flags[DivisionByZero])
 
+    def test_divide_unbound_context(self):
+        with self.unbound_context() as c:
+            x = self.decimal.Decimal('1')
+            y = x // 1  # should be fast
+
+    def test_remainder_near(self):
+        L = 1000
+        limit = sys.get_int_max_str_digits()
+        sys.set_int_max_str_digits(L)
+        self.addCleanup(sys.set_int_max_str_digits, limit)
+
+        with self.unbound_context(prec=2 * L) as c:
+            self.assertEqual(c.prec, 2 * L)
+            x = self.decimal.Decimal(f'1e{L}')
+            y = x.remainder_near(1)  # must not raise a ValueError
+
     def test_power(self):
         Decimal = self.decimal.Decimal
         localcontext = self.decimal.localcontext
diff --git a/Misc/NEWS.d/next/Library/2025-10-13-15-43-09.gh-issue-140036.b_59uN.rst b/Misc/NEWS.d/next/Library/2025-10-13-15-43-09.gh-issue-140036.b_59uN.rst
new file mode 100644
index 00000000000000..4bd92b4d735225
--- /dev/null
+++ b/Misc/NEWS.d/next/Library/2025-10-13-15-43-09.gh-issue-140036.b_59uN.rst
@@ -0,0 +1,2 @@
+Avoid hanging in floor division of pure Python :class:`decimal.Decimal`
+instances when the context precision is very large. Patch by Bénédikt Tran.