diff --git a/llvm/lib/Analysis/InstructionSimplify.cpp b/llvm/lib/Analysis/InstructionSimplify.cpp
--- a/llvm/lib/Analysis/InstructionSimplify.cpp
+++ b/llvm/lib/Analysis/InstructionSimplify.cpp
@@ -1079,6 +1079,16 @@
   }
 
   // IsSigned == false.
+
+  // Is the unsigned dividend known to be less than a constant divisor?
+  // TODO: Convert this (and above) to range analysis
+  //      ("computeConstantRangeIncludingKnownBits")?
+  const APInt *C;
+  if (match(Y, m_APInt(C)) &&
+      computeKnownBits(X, Q.DL, 0, Q.AC, Q.CxtI, Q.DT).getMaxValue().ult(*C))
+    return true;
+
+  // Try again for any divisor:
   // Is the dividend unsigned less than the divisor?
   return isICmpTrue(ICmpInst::ICMP_ULT, X, Y, Q, MaxRecurse);
 }
diff --git a/llvm/test/Transforms/InstSimplify/div.ll b/llvm/test/Transforms/InstSimplify/div.ll
--- a/llvm/test/Transforms/InstSimplify/div.ll
+++ b/llvm/test/Transforms/InstSimplify/div.ll
@@ -155,10 +155,7 @@
 
 define i8 @udiv_dividend_known_smaller_than_constant_divisor2(i1 %b) {
 ; CHECK-LABEL: @udiv_dividend_known_smaller_than_constant_divisor2(
-; CHECK-NEXT:    [[T0:%.*]] = zext i1 [[B:%.*]] to i8
-; CHECK-NEXT:    [[XOR:%.*]] = xor i8 [[T0]], 12
-; CHECK-NEXT:    [[R:%.*]] = udiv i8 [[XOR]], 14
-; CHECK-NEXT:    ret i8 [[R]]
+; CHECK-NEXT:    ret i8 0
 ;
   %t0 = zext i1 %b to i8
   %xor = xor i8 %t0, 12
@@ -166,6 +163,8 @@
   ret i8 %r
 }
 
+; negative test - dividend can equal 13
+
 define i8 @not_udiv_dividend_known_smaller_than_constant_divisor2(i1 %b) {
 ; CHECK-LABEL: @not_udiv_dividend_known_smaller_than_constant_divisor2(
 ; CHECK-NEXT:    [[T0:%.*]] = zext i1 [[B:%.*]] to i8
diff --git a/llvm/test/Transforms/InstSimplify/rem.ll b/llvm/test/Transforms/InstSimplify/rem.ll
--- a/llvm/test/Transforms/InstSimplify/rem.ll
+++ b/llvm/test/Transforms/InstSimplify/rem.ll
@@ -187,8 +187,7 @@
 ; CHECK-LABEL: @urem_dividend_known_smaller_than_constant_divisor2(
 ; CHECK-NEXT:    [[T0:%.*]] = zext i1 [[B:%.*]] to i8
 ; CHECK-NEXT:    [[XOR:%.*]] = xor i8 [[T0]], 12
-; CHECK-NEXT:    [[R:%.*]] = urem i8 [[XOR]], 14
-; CHECK-NEXT:    ret i8 [[R]]
+; CHECK-NEXT:    ret i8 [[XOR]]
 ;
   %t0 = zext i1 %b to i8
   %xor = xor i8 %t0, 12
@@ -196,6 +195,8 @@
   ret i8 %r
 }
 
+; negative test - dividend can equal 13
+
 define i8 @not_urem_dividend_known_smaller_than_constant_divisor2(i1 %b) {
 ; CHECK-LABEL: @not_urem_dividend_known_smaller_than_constant_divisor2(
 ; CHECK-NEXT:    [[T0:%.*]] = zext i1 [[B:%.*]] to i8