Smallest Good Base - Practice Coding Problems

Smallest Good Base - Problem

You're given a positive integer n represented as a string, and your task is to find the smallest good base of n.

What makes a base "good"? A base k (where k ≥ 2) is considered good for number n if when we convert n to base k, all digits are 1's.

For example, if n = 13, then base k = 3 is good because 13 in base 3 is "111" (since 1×3² + 1×3¹ + 1×3⁰ = 9 + 3 + 1 = 13).

Goal: Return the smallest such base k as a string.

Input & Output

example_1.py — Basic case

$ Input: n = "13"

› Output: "3"

💡 Note: 13 in base 3 is "111" because 1×3² + 1×3¹ + 1×3⁰ = 9 + 3 + 1 = 13. Base 3 is the smallest base where all digits are 1.

example_2.py — Larger number

$ Input: n = "4681"

› Output: "8"

💡 Note: 4681 in base 8 is "11111" because 1×8⁴ + 1×8³ + 1×8² + 1×8¹ + 1×8⁰ = 4096 + 512 + 64 + 8 + 1 = 4681.

example_3.py — Edge case

$ Input: n = "3"

› Output: "2"

💡 Note: 3 in base 2 is "11" because 1×2¹ + 1×2⁰ = 2 + 1 = 3. Base 2 is the smallest good base.

Constraints

n is represented as a string
1 ≤ n ≤ 10¹⁸
n is guaranteed to have at least one good base
The answer will fit in a 64-bit signed integer

Visualization

Tap to expand

Understanding the Visualization

Identify the pattern

A number with all 1's in base k follows the pattern: 1 + k + k² + k³ + ...

Use geometric series formula

This sum equals (k^m - 1)/(k - 1) where m is the number of digits

Binary search optimization

For each possible m, binary search for k that satisfies the equation

Key Takeaway

🎯 Key Insight: Transform the digit-checking problem into a mathematical equation, then use binary search to efficiently find the solution in O(log²n) time instead of O(n).

Asked in

G Google 35 f Facebook 25 a Amazon 20 ⊞ Microsoft 15

The optimal solution uses mathematical insight combined with binary search. Key insight: if n has m digits in base k (all 1's), then n = (k^m - 1)/(k - 1). We iterate through possible values of m and binary search for the corresponding k, achieving O(log²n) time complexity.

Common Approaches

Approach	Time	Space	Notes
✓ Brute Force (Try Every Base)	O(n * log n)	O(1)	Try every possible base from 2 to n-1 and check if all digits are 1's
Mathematical Binary Search (Optimal)	O(log²n)	O(1)	Use mathematical insight that n = (k^m - 1)/(k - 1) for m digits, then binary search for k

Brute Force (Try Every Base) — Algorithm Steps

Start with base k = 2
For current base k, check if n can be written as sum of powers of k with all coefficients being 1
If yes, return k as it's the smallest good base
If no, increment k and repeat
Continue until we find a good base

Visualization

Tap to expand

Step-by-Step Walkthrough

Start with base 2

Test if n can be written as sum of powers of 2 with all coefficients being 1

Check validity

Divide n by current base repeatedly, checking if remainder is always 1

Move to next base

If current base doesn't work, increment and test the next base

Code -

solution.c — C

#include <stdio.h>
#include <stdlib.h>
#include <string.h>

char* smallestGoodBase(char* n) {
    long long num = atoll(n);
    char* result = malloc(20);
    
    for (long long base = 2; base < num; base++) {
        long long temp = num;
        int valid = 1;
        
        while (temp > 0) {
            if (temp % base != 1) {
                valid = 0;
                break;
            }
            temp /= base;
        }
        
        if (valid && temp == 0) {
            sprintf(result, "%lld", base);
            return result;
        }
    }
    
    sprintf(result, "%lld", num - 1);
    return result;
}

int main() {
    char n[20];
    scanf("%s", n);
    // Remove quotes if present
    if (n[0] == '"') {
        memmove(n, n + 1, strlen(n));
        n[strlen(n) - 1] = '\0';
    }
    printf("%s\n", smallestGoodBase(n));
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(n * log n)

We try O(n) bases, and for each base we do O(log n) operations to check if it's valid

⚡ Linearithmic

Space Complexity

O(1)

Only using a few variables to track the current base and calculations

✓ Linear Space

28.5K Views

Medium Frequency

~25 min Avg. Time

842 Likes

Ln 1, Col 1

Smart Actions

💡 Explanation

AI Ready

💡 Suggestion Tab to accept Esc to dismiss

// Output will appear here after running code

Code Editor Closed

Click the red button to reopen

Input & Output

Constraints

Visualization

Related Problems

Common Approaches

Brute Force (Try Every Base) — Algorithm Steps

Visualization

Code -

Time & Space Complexity

Select Compiler