diff --git a/stdlib/Random/src/RNGs.jl b/stdlib/Random/src/RNGs.jl
index 281acad533dad..e9b7f152ef4fe 100644
--- a/stdlib/Random/src/RNGs.jl
+++ b/stdlib/Random/src/RNGs.jl
@@ -232,12 +232,11 @@ function mt_pop!(r::MersenneTwister, ::Type{T}) where T<:BitInteger
     (x128 >> (i128 * (sizeof(T) << 3))) % T
 end
 
-# not necessary, but very slightly more efficient
 function mt_pop!(r::MersenneTwister, ::Type{T}) where {T<:Union{Int128,UInt128}}
     reserve1(r, T)
-    @inbounds res = r.ints[r.idxI >> 4]
-    r.idxI -= 16
-    res % T
+    idx = r.idxI >> 4
+    r.idxI = idx << 4 - 16
+    @inbounds r.ints[idx] % T
 end
 
 
diff --git a/stdlib/Random/test/runtests.jl b/stdlib/Random/test/runtests.jl
index b78e3ae4b8a1f..2aeea0f623877 100644
--- a/stdlib/Random/test/runtests.jl
+++ b/stdlib/Random/test/runtests.jl
@@ -801,3 +801,35 @@ end
 @testset "RNGs broadcast as scalars: T" for T in (MersenneTwister, RandomDevice)
     @test length.(rand.(T(), 1:3)) == 1:3
 end
+
+@testset "generated scalar integers do not overlap" begin
+    m = MersenneTwister()
+    xs = reinterpret(UInt64, m.ints)
+    x = rand(m, UInt128)  # m.idxI % 16 == 0
+    @test x % UInt64 == xs[end-1]
+    x = rand(m, UInt64)
+    @test x == xs[end-2]
+    x = rand(m, UInt64)
+    @test x == xs[end-3]
+    x = rand(m, UInt64)
+    @test x == xs[end-4]
+    x = rand(m, UInt128) # m.idxI % 16 == 8
+    @test (x >> 64) % UInt64 == xs[end-6]
+    @test x % UInt64 == xs[end-7]
+    x = rand(m, UInt64)
+    @test x == xs[end-8] # should not be == xs[end-7]
+
+    s = Set{UInt64}()
+    n = 0
+    for _=1:2000
+        x = rand(m, rand((UInt64, UInt128, Int64, Int128)))
+        if sizeof(x) == 8
+            push!(s, x % UInt64)
+            n += 1
+        else
+            push!(s, x % UInt64, (x >> 64) % UInt64)
+            n += 2
+        end
+    end
+    @test length(s) == n
+end