diff --git a/src/ModulusRemainder.h b/src/ModulusRemainder.h
index c0341b75abf6..cbcdce10b98c 100644
--- a/src/ModulusRemainder.h
+++ b/src/ModulusRemainder.h
@@ -7,6 +7,8 @@
 
 #include <cstdint>
 
+#include "Util.h"
+
 namespace Halide {
 
 struct Expr;
@@ -83,8 +85,8 @@ ModulusRemainder modulus_remainder(const Expr &e, const Scope<ModulusRemainder>
 /** Reduce an expression modulo some integer. Returns true and assigns
  * to remainder if an answer could be found. */
 ///@{
-bool reduce_expr_modulo(const Expr &e, int64_t modulus, int64_t *remainder);
-bool reduce_expr_modulo(const Expr &e, int64_t modulus, int64_t *remainder, const Scope<ModulusRemainder> &scope);
+HALIDE_MUST_USE_RESULT bool reduce_expr_modulo(const Expr &e, int64_t modulus, int64_t *remainder);
+HALIDE_MUST_USE_RESULT bool reduce_expr_modulo(const Expr &e, int64_t modulus, int64_t *remainder, const Scope<ModulusRemainder> &scope);
 ///@}
 
 void modulus_remainder_test();
diff --git a/src/Solve.cpp b/src/Solve.cpp
index a08eedadbd27..22bd14e44412 100644
--- a/src/Solve.cpp
+++ b/src/Solve.cpp
@@ -44,18 +44,22 @@ class SolveExpression : public IRMutator {
         map<Expr, CacheEntry, ExprCompare>::iterator iter = cache.find(e);
         if (iter == cache.end()) {
             // Not in the cache, call the base class version.
-            debug(4) << "Mutating " << e << " (" << uses_var << ")\n";
+            debug(4) << "Mutating " << e << " (" << uses_var << ", " << failed << ")\n";
             bool old_uses_var = uses_var;
             uses_var = false;
+            bool old_failed = failed;
+            failed = false;
             Expr new_e = IRMutator::mutate(e);
-            CacheEntry entry = {new_e, uses_var};
+            CacheEntry entry = {new_e, uses_var, failed};
             uses_var = old_uses_var || uses_var;
+            failed = old_failed || failed;
             cache[e] = entry;
-            debug(4) << "(Miss) Rewrote " << e << " -> " << new_e << " (" << uses_var << ")\n";
+            debug(4) << "(Miss) Rewrote " << e << " -> " << new_e << " (" << uses_var << ", " << failed << ")\n";
             return new_e;
         } else {
             // Cache hit.
             uses_var = uses_var || iter->second.uses_var;
+            failed = failed || iter->second.failed;
             debug(4) << "(Hit) Rewrote " << e << " -> " << iter->second.expr << " (" << uses_var << ")\n";
             return iter->second.expr;
         }
@@ -75,7 +79,7 @@ class SolveExpression : public IRMutator {
     // stateless, so we can cache everything.
     struct CacheEntry {
         Expr expr;
-        bool uses_var;
+        bool uses_var, failed;
     };
     map<Expr, CacheEntry, ExprCompare> cache;
 
@@ -388,16 +392,25 @@ class SolveExpression : public IRMutator {
         const Mul *mul_a = a.as<Mul>();
         Expr expr;
         if (a_uses_var && !b_uses_var) {
+            const int64_t *ib = as_const_int(b);
+            auto is_multiple_of_b = [&](const Expr &e) {
+                if (ib) {
+                    int64_t r = 0;
+                    return reduce_expr_modulo(e, *ib, &r) && r == 0;
+                } else {
+                    return can_prove(e / b * b == e);
+                }
+            };
             if (add_a && !a_failed &&
-                can_prove(add_a->a / b * b == add_a->a)) {
+                is_multiple_of_b(add_a->a)) {
                 // (f(x) + a) / b -> f(x) / b + a / b
                 expr = mutate(simplify(add_a->a / b) + add_a->b / b);
             } else if (sub_a && !a_failed &&
-                       can_prove(sub_a->a / b * b == sub_a->a)) {
+                       is_multiple_of_b(sub_a->a)) {
                 // (f(x) - a) / b -> f(x) / b - a / b
                 expr = mutate(simplify(sub_a->a / b) - sub_a->b / b);
             } else if (mul_a && !a_failed && no_overflow_int(op->type) &&
-                       can_prove(mul_a->b / b * b == mul_a->b)) {
+                       is_multiple_of_b(mul_a->b)) {
                 // (f(x) * a) / b -> f(x) * (a / b)
                 expr = mutate(mul_a->a * (mul_a->b / b));
             }
@@ -776,6 +789,7 @@ class SolveExpression : public IRMutator {
         } else if (scope.contains(op->name)) {
             CacheEntry e = scope.get(op->name);
             uses_var = uses_var || e.uses_var;
+            failed = failed || e.failed;
             return e.expr;
         } else if (external_scope.contains(op->name)) {
             Expr e = external_scope.get(op->name);
@@ -790,11 +804,14 @@ class SolveExpression : public IRMutator {
 
     Expr visit(const Let *op) override {
         bool old_uses_var = uses_var;
+        bool old_failed = failed;
         uses_var = false;
+        failed = false;
         Expr value = mutate(op->value);
-        CacheEntry e = {value, uses_var};
-
+        CacheEntry e = {value, uses_var, failed};
         uses_var = old_uses_var;
+        failed = old_failed;
+
         ScopedBinding<CacheEntry> bind(scope, op->name, e);
         return mutate(op->body);
     }