Smallest All-Ones Multiple — Python

# Time:  O(k)
# Space: O(1)

# math, leetcode #1015
class Solution(object):
    def minAllOneMultiple(self, k):
        """
        :type k: int
        :rtype: int
        """
        # by observation, k % 2 = 0 or k % 5 = 0, it is impossible
        if k % 2 == 0 or k % 5 == 0:
            return -1

        # let f(n) is a n-length integer only containing digit 1
        # if there is no n in range (1..k) s.t. f(n) % k = 0
        # => there must be k remainders of f(n) % k in range (1..k-1) excluding 0
        # => due to pigeonhole principle, there must be at least 2 same remainders
        # => there must be some x, y in range (1..k) and x > y s.t. f(x) % k = f(y) % k
        # => (f(x) - f(y)) % k = 0
        # => (f(x-y) * 10^y) % k = 0
        # => due to (x-y) in range (1..k)
        # => f(x-y) % k != 0
        # => 10^y % k = 0
        # => k % 2 = 0 or k % 5 = 0
        # => -><-
        # it proves that there must be some n in range (1..k) s.t. f(n) % k = 0
        r = 0
        for n in xrange(1, k+1):
            r = (r*10+1)%k
            if not r:
                return n
        assert(False)
        return -1  # never reach