Dyalog APL, 25

This defines a proper program "P" (not just an unnamed function):

P←2∘{0::⍵∇⍨1+⍺⋄⍺⊣~⍎¨⍕⍺×⍵} 

2∘ begin with 2 as left argument
0:: if there is any error...
⍵∇⍨1+⍺ call itself with an incremented left argument
⍺×⍵ multiply left and right arguments
⍕ make into string
⍎¨ make each character into a number
~ attempt logical NOT (if it fails, go to error-handling above, else...)
⍺⊣ return the current left argument.

 P 2 50 P 21 481 P¨⍳8 ⍝ 1 through 8 10 5 37 25 2 185 143 125 

Jelly, 8 bytes

×QḌ=⁵ð1# 

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Tied Pyth!

How it works

×QḌ=⁵ð1# - Main link. Takes n on the left ð - Combine the previous links into a dyad f(i, n): × - i×n Q - Unique digits Ḍ - Convert into an integer =⁵ - Does this equal 10? f(i, n) returns true for a given i if i×n only consists of 1 and 0 1# - Starting from n, find the first integer i = n, n+1, ... such that f(i, n) is true 

Vyxal, 13 bytes

∞›*'S₀S↔=;h$/ 

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∞›* # All multpiles of n ' ; # Filtered by S₀S↔ # When you remove all the digits except 0 and 1 = # Does it equal itself? h # Get the first item for which this happens $/ # Divide by n 

Vyxal, 82 bits^v2, 10.25 bytes

¨*'₀τGċ¬;h?/ 

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Vyxal, 44 bits^v2, 5.5 bytes

?*↑ṅ)ṅ 

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Thunno 2, 4 bytes

Ƙ×Gḅ 

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Explanation

Ƙ×Gḅ # Implicit input Ƙ # First integer where: × # Multiply by the input G # Maximum digit ḅ # Equals one? # Implicit output 

PowerShell Core, 35 bytes

for(;++$i*"$args"-replace"0|1"){}$i 

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C#, 156+31=187 bytes

The code requires the following usings:

using System;using System.Linq; 

Here the code:

class P{static void Main(string[]a){var b=ulong.Parse(Console.ReadLine());var d=1U;for(;(b*d).ToString().Any(x=>x!='0'&&x!='1');d++);Console.WriteLine(d);}} 

Ungolfed:

using System; using System.Linq; class P { static void Main(string[] a) { var b = ulong.Parse(Console.ReadLine()); var d = 1U; for (; (b*d).ToString().Any(x => x != '0' && x != '1'); d++); Console.WriteLine(d); } } 

SAS, 81 bytes

This is give or take the same approach as some of the 'fast' python approaches above: iterate through the possible binary numbers, pretend they're decimal, test them.

data a;b=72;do until(a=int(a));y+1;a=input(put(y,binary32.),32.)/b;end;put a;run; 

Unfortunately I can't skip the input/put, because the default width for implicit conversions is only 12 (meaning it quickly becomes "1e8" and similar which fails).

Ungolfed:

data a; b=72; do until(a=int(a)); y+1; a=input(put(y,binary32.),32.)/b; end; put a; run; 

This isn't entirely safe; for larger numbers (which have no solution in a double precision floating point environment) this could turn into an infinite loop.

Ruby 46 bytes

No better in # of bytes than the previous Ruby solution by the Peter Lenkefi, but slightly different approach:

gets;p 1.step.find{|i|"#{$_.to_i*i}"!~/[^01]/} 

Burlesque, 24 bytes

Found a shorter version:

1R@?*b0{{"10"j~[}al}fi?i 

Older Versions:

1R@?*b0{"\`[01]*\\'"~=}fi?i 

It's terribly slow though if the number can't be found fast.

1R@?*b0{"23456789"INnu}fi?i 

This version might be faster because it doesn't use regexes. Still terribly slow though.

There's no limit except the time it takes to find the answer which can be quite long.

Husk, 9 bytes

ḟö=1▲d*⁰N 

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ḟö=1▲d*⁰N ḟö # first truthy element of: *⁰N # multiply all natural numbers by input ▲d # max of decimal digits =1 # equals 1 

Japt, 13 12 10 bytes

È*U skA}f1 

Try it - Be careful with multiples of 9 ;)

Ruby, 36 bytes

->n{(1..).find{(_1*n).digits.max<2}} 

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