diff --git a/src/sage/rings/integer.pyx b/src/sage/rings/integer.pyx
index 4c835d0..77a3a18 100644
--- a/src/sage/rings/integer.pyx
+++ b/src/sage/rings/integer.pyx
@@ -4414,6 +4414,29 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
         sig_off()
         return z
 
+    def _gcd(self, Integer n):
+        """
+        Return the greatest common divisor of self and `n`.
+
+        EXAMPLES::
+
+            sage: 1._gcd(-1)
+            1
+            sage: 0._gcd(1)
+            1
+            sage: 0._gcd(0)
+            0
+            sage: 2._gcd(2^6)
+            2
+            sage: 21._gcd(2^6)
+            1
+        """
+        cdef Integer z = PY_NEW(Integer)
+        sig_on()
+        mpz_gcd(z.value, self.value, n.value)
+        sig_off()
+        return z
+
     def denominator(self):
         """
         Return the denominator of this integer, which of course is
@@ -6736,33 +6759,6 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
             raise ZeroDivisionError(f"inverse of Mod({self}, {m}) does not exist")
         return ans
 
-    def gcd(self, n):
-        """
-        Return the greatest common divisor of self and `n`.
-
-        EXAMPLES::
-
-            sage: gcd(-1,1)
-            1
-            sage: gcd(0,1)
-            1
-            sage: gcd(0,0)
-            0
-            sage: gcd(2,2^6)
-            2
-            sage: gcd(21,2^6)
-            1
-        """
-        if not isinstance(n, Integer) and not isinstance(n, int):
-            left, right = coercion_model.canonical_coercion(self, n)
-            return left.gcd(right)
-        cdef Integer m = as_Integer(n)
-        cdef Integer g = PY_NEW(Integer)
-        sig_on()
-        mpz_gcd(g.value, self.value, m.value)
-        sig_off()
-        return g
-
     def crt(self, y, m, n):
         """
         Return the unique integer between `0` and `mn` that is congruent to
diff --git a/src/sage/structure/element.pyx b/src/sage/structure/element.pyx
index f145e66..1cf634e 100644
--- a/src/sage/structure/element.pyx
+++ b/src/sage/structure/element.pyx
@@ -3908,16 +3908,79 @@ def is_PrincipalIdealDomainElement(x):
     return isinstance(x, PrincipalIdealDomainElement)
 
 cdef class PrincipalIdealDomainElement(DedekindDomainElement):
+    def gcd(self, right):
+        r"""
+        Return the greatest common divisor of ``self`` and ``other``.
+
+        TESTS:
+
+        :trac:`30849`::
+
+            sage: 2.gcd(pari(3))
+            1
+            sage: type(2.gcd(pari(3)))
+            <class 'sage.rings.integer.Integer'>
+
+            sage: 2.gcd(pari('1/3'))
+            1/3
+            sage: type(2.gcd(pari('1/3')))
+            <class 'sage.rings.rational.Rational'>
+
+            sage: import gmpy2
+            sage: 2.gcd(gmpy2.mpz(3))
+            1
+            sage: type(2.gcd(gmpy2.mpz(3)))
+            <class 'sage.rings.integer.Integer'>
+
+            sage: 2.gcd(gmpy2.mpq(1,3))
+            1/3
+            sage: type(2.gcd(pari('1/3')))
+            <class 'sage.rings.rational.Rational'>
+        """
+        # NOTE: in order to handle nicely pari or gmpy2 integers we do not rely only on coercion
+        if not isinstance(right, Element):
+            right = py_scalar_to_element(right)
+            if not isinstance(right, Element):
+                right = right.sage()
+        if not ((<Element>right)._parent is self._parent):
+            from sage.arith.all import gcd
+            return coercion_model.bin_op(self, right, gcd)
+        return self._gcd(right)
+
     def lcm(self, right):
         """
         Return the least common multiple of ``self`` and ``right``.
-        """
-        if not isinstance(right, Element) or not ((<Element>right)._parent is self._parent):
+
+        TESTS:
+
+        :trac:`30849`::
+
+            sage: 2.lcm(pari(3))
+            6
+            sage: type(2.lcm(pari(3)))
+            <class 'sage.rings.integer.Integer'>
+
+            sage: 2.lcm(pari('1/3'))
+            2
+            sage: type(2.lcm(pari('1/3')))
+            <class 'sage.rings.rational.Rational'>
+
+            sage: import gmpy2
+            sage: 2.lcm(gmpy2.mpz(3))
+            6
+            sage: type(2.lcm(gmpy2.mpz(3)))
+            <class 'sage.rings.integer.Integer'>
+        """
+        # NOTE: in order to handle nicely pari or gmpy2 integers we do not rely only on coercion
+        if not isinstance(right, Element):
+            right = py_scalar_to_element(right)
+            if not isinstance(right, Element):
+                right = right.sage()
+        if not ((<Element>right)._parent is self._parent):
             from sage.arith.all import lcm
             return coercion_model.bin_op(self, right, lcm)
         return self._lcm(right)
 
-
 # This is pretty nasty low level stuff. The idea is to speed up construction
 # of EuclideanDomainElements (in particular Integers) by skipping some tp_new
 # calls up the inheritance tree.