diff --git a/src/sage/rings/integer.pyx b/src/sage/rings/integer.pyx
index 7d0e6f34464..1b4d3a1ff10 100644
--- a/src/sage/rings/integer.pyx
+++ b/src/sage/rings/integer.pyx
@@ -6975,11 +6975,9 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
 
     def crt(self, y, m, n):
         """
-        Return the unique integer between `0` and `mn` that is congruent to
+        Return the unique integer between `0` and `\\lcm(m,n)` that is congruent to
         the integer modulo `m` and to `y` modulo `n`.
 
-        We assume that `m` and `n` are coprime.
-
         EXAMPLES::
 
             sage: n = 17
@@ -6989,6 +6987,16 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
             17
             sage: m%11
             5
+
+        ``crt`` also works for some non-coprime moduli::
+
+            sage: 6.crt(0,10,4)
+            16
+            sage: 6.crt(0,10,10)
+            Traceback (most recent call last):
+            ...
+            ValueError: no solution to crt problem since gcd(10,10) does not
+            divide 6 - 0
         """
         cdef object g, s
         cdef Integer _y, _m, _n
@@ -6997,7 +7005,9 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
         _n = Integer(n)
         g, s, _ = _m.xgcd(_n)
         if not g.is_one():
-            raise ArithmeticError("CRT requires that gcd of moduli is 1.")
+            if (self % g) != (_y % g):
+                raise ValueError(f"no solution to crt problem since gcd({_m},{_n}) does not divide {self} - {_y}")
+            return (self + g * Integer(0).crt((_y - self) // g, _m // g, _n // g)) % _m.lcm(_n)
         # Now s*m + t*n = 1, so the answer is x + (y-x)*s*m, where x=self.
         return (self + (_y - self) * s * _m) % (_m * _n)