Given n > 1 there is an m > 1 such that nm has only 0s and 1s base 10

Needs["MultivariateStatistics`"]
Needs["Histograms`"]
Needs["BarCharts`"]

FromDigits[IntegerDigits[1000000,2]]
FromDigits[IntegerDigits[110000,2]]
23886/(60 60.)

(*list = {1,10,11,100,101,110,111,1000,1001,1010,1011,1100,1101,1110,1111};*)
list = {};
Timing[For[x= 1, x ≤1100000, x++,
{
list = AppendTo[list,FromDigits[IntegerDigits[x,2],10]];
}];]
Print[list[[Length[list]]]]

numfound = 0;
end = 10000;
start = 1;
badones = {};
successlist = {};
print = 0;
For[n = start, n ≤end, n++,
{
found = 0;
If[ Mod[10 n,(end - start+1)] == 0, Print["We've done ", 100.0 n / (end-start+1), "%."]];
For[j = 1, j ≤ Length[list], j++,
{
If[Mod[list[[j]],n] == 0, 
{
If[print == 1,Print["n = ", n, " and number is ", list[[j]]]];
numfound = numfound + 1;
found = 1;
successlist = AppendTo[successlist, Log[10,1.0 list[[j]]]];
j = Length[list] + 10;
} ]; (* end of IF *)
}]; (* end of FOR *)
If[found == 0, 
{
If[print == 1,Print["Didn't Find Anything that worked!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"]];
badlones = AppendTo[badones, n];
}];
}];
Print[" "];
Print["Of ", end - start  + 1, " we found ", numfound, " which is ", 100.  numfound / (end - start+1) , "%."];
Print["Bad ones are ", badones];
Print[Histogram[successlist, HistogramScale -> 1]];
Print["Histogram of log of the number with all 0s and 1s with n as a factor."];
Print["Range is from ", start, " to ", end, "."];
Print["Couldn't find a factor for ",badones, "."];
Print["Note: only looking at numbers up to ", list[[Length[list]]], " which is of size 10^", Log[10,1.0 list[[Length[list]]]], "."];