diff --git a/bignum.c b/bignum.c
index fe22aaeb..942e7200 100644
--- a/bignum.c
+++ b/bignum.c
@@ -480,7 +480,8 @@ sexp sexp_bignum_quot_rem (sexp ctx, sexp *rem, sexp a, sexp b) {
   sexp_bignum_sign(b1) = 1;
   q = SEXP_ZERO;
   x = sexp_make_bignum(ctx, sexp_bignum_length(a));
-  while (sexp_bignum_compare(a1, b1) > 0) {
+  while (sexp_bignum_compare_abs(a1, b1) > 0) {
+    /* guess divisor x */
     alen = sexp_bignum_hi(a1);
     d = sexp_bignum_data(a1)[alen-1] / sexp_bignum_data(b1)[blen-1];
     memset(sexp_bignum_data(x), 0, sexp_bignum_length(x)*sizeof(sexp_uint_t));
@@ -491,14 +492,26 @@ sexp sexp_bignum_quot_rem (sexp ctx, sexp *rem, sexp a, sexp b) {
     } else {
       sexp_bignum_data(x)[alen - blen] = d;
     }
+    /* update quotient q and remainder a1 estimates */
     y = sexp_bignum_mul(ctx, NULL, b1, x);
-    a1 = sexp_bignum_sub(ctx, NULL, a1, y);
-    q = (sign < 0) ? sexp_sub(ctx, q, x) : sexp_add(ctx, q, x);
-    if (sexp_bignum_sign(a1) < 0) {
-      sexp_bignum_sign(a1) = 1;
+    if (sign < 0) {
+      a1 = sexp_bignum_add(ctx, NULL, a1, y);
+      q = sexp_sub(ctx, q, x);
+    } else {
+      a1 = sexp_bignum_sub(ctx, NULL, a1, y);
+      q = sexp_add(ctx, q, x);
+    }
+    /* flip the sign if we overshot in our estimate */
+    if (sexp_bignum_sign(a1) != sign) {
+      sexp_bignum_sign(a1) = -sign;
       sign *= -1;
     }
   }
+  /* adjust signs */
+  if (sign < 0) {
+    q = sexp_sub(ctx, q, SEXP_ONE);
+    a1 = sexp_add(ctx, a1, b1);
+  }
   if (sexp_bignum_sign(a) * sexp_bignum_sign(b) < 0) {
     sexp_negate_exact(q);
   }
diff --git a/tests/numeric-tests.scm b/tests/numeric-tests.scm
index 80362b9c..a89b0c5f 100644
--- a/tests/numeric-tests.scm
+++ b/tests/numeric-tests.scm
@@ -176,6 +176,10 @@
   (test 1.0 (max 1/2 1.0))
   (test 18446744073709551617 (numerator (/ 18446744073709551617 2)))
   (test "18446744073709551617/2" (number->string (/ 18446744073709551617 2)))
+  (let ((a 1000000000000000000000000000000000000000)
+        (b 31622776601683794000))
+    (test 31622776601683792639 (quotient a b))
+    (test 30922992657207634000 (remainder a b)))
   (let ((r  (/ (expt 2 61) 3)))
     (test 0 (- r r))
     (test 2305843009213693952/3 r)))
diff --git a/tests/r7rs-tests.scm b/tests/r7rs-tests.scm
index 2fa8311a..2a3bf916 100644
--- a/tests/r7rs-tests.scm
+++ b/tests/r7rs-tests.scm
@@ -224,6 +224,10 @@
     (let*-values (((root rem) (exact-integer-sqrt (expt 2 121))))
       (list root rem)))
 
+(test '(31622776601683793319 62545769258890964239)
+    (let*-values (((root rem) (exact-integer-sqrt (expt 10 39))))
+      (list root rem)))
+
 (let*-values (((root rem) (exact-integer-sqrt (expt 2 140))))
   (test 0 rem)
   (test (expt 2 140) (square root)))