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

There's an O(n)-time (arithmetic operations mod n, really) solution, which is more efficient than the answer currently accepted. The idea is to construct a graph on vertices 0..n-1 where vertex i has connections to (i*10)%n and (i*10+1)%n, then use breadth-first search to find the lexicographically least path from 1 to 0.

def smallest(n):
    parents = {}
    queue = [(1 % n, 1, None)]
    i = 0
    while i < len(queue):
        residue, digit, parent = queue[i]
        i += 1
        if residue in parents:
            continue
        if residue == 0:
            answer = []
            while True:
                answer.append(str(digit))
                if parent is None:
                    answer.reverse()
                    return ''.join(answer)
                digit, parent = parents[parent]
        parents[residue] = (digit, parent)
        for digit in (0, 1):
            queue.append(((residue * 10 + digit) % n, digit, residue))
    return None