How to find the smallest number with just 0 and 1 which is divided by a given number?

Question

Every positive integer divide some number whose representation (base 10) contains only zeroes and ones.

One can prove that:

Consider the numbers 1, 11, 111,

Answer 1

This is a fast way to get the first 792 answers. Def the most simple code:

__author__ = 'robert'

from itertools import product

def get_nums(max_length):
    assert max_length < 21 #Otherwise there will be over 2 million possibilities
    for length in range(1, max_length + 1):
        for prod in product("10", repeat=length):
            if prod[0] == '1':
                yield int("".join(prod))

print list(get_nums(4))
[1, 11, 10, 111, 110, 101, 100, 1111, 1110, 1101, 1100, 1011, 1010, 1001, 1000]

nums = sorted(get_nums(20))
print len(nums)

solution = {}

operations = 0

for factor in range(1, 1000):
    for num in nums:
        operations += 1
        if num % factor == 0:
            solution[factor] = num
            break
    print factor, operations
    if factor not in solution:
        print "no solution for factor %s" % factor
        break

print solution[787]

max_v = max(solution.values())
for factor, val in solution.items():
    if val == max_v:
        print factor, max_v


[1, 11, 10, 111, 110, 101, 100, 1111, 1110, 1101, 1100, 1011, 1010, 1001, 1000]
1048575
1 1
2 3
3 10
4 14
5 16
6 30
7 39
8 47
9 558
10 560
11 563
12 591
13 600
14 618
15 632
16 648
17 677
18 1699
19 1724
20 1728
..
..
187 319781
188 319857
..
..
791 4899691
792 5948266
no solution for factor 792
10110001111
396 11111111111111111100