README.md

483. Smallest Good Base

Given an integer n represented as a string, return the smallest good base of n.

We call k >= 2 a good base of n, if all digits of n base k are 1's.

Example 1:

Input: n = "13"
Output: "3"
Explanation: 13 base 3 is 111.

Example 2:

Input: n = "4681"
Output: "8"
Explanation: 4681 base 8 is 11111.

Example 3:

Input: n = "1000000000000000000"
Output: "999999999999999999"
Explanation: 1000000000000000000 base 999999999999999999 is 11.

Constraints:

• n is an integer in the range [3, 1018].
• n does not contain any leading zeros.

Solutions (Python)

1. Solution

import math


class Solution:
    def smallestGoodBase(self, n: str) -> str:
        n = int(n)

        for x in range(math.ceil(math.log2(n)), 1, -1):
            l, r = 2, n - 1

            while l <= r:
                k = (l + r) // 2

                if k ** x - 1 == n * (k - 1):
                    return str(k)
                elif k ** x - 1 < n * (k - 1):
                    l = k + 1
                else:
                    r = k - 1