EP Ethan Pham Project Euler, competitive programming, and notes on the things I keep studying.
ICPC 2016
ICPC 2016

E. Forever Young

My birthday is coming up. Alas, I am getting old and would like to feel young again. Fortunately, I have come up with an excellent way of feeling younger: if I write my age as a number in an appropriately chosen base b, then it appears to be smaller. For in...

Updated May 21, 2026
Track ICPC
Year 2016
Statement Text + PDF
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Problem statement

Problem Statement

Formatted from the contest statement text, with sample tests broken out into copyable blocks.

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Time limit 1 second

My birthday is coming up. Alas, I am getting old and would like to feel young again. Fortunately, I have come up with an excellent way of feeling younger: if I write my age as a number in an appropriately chosen base b, then it appears to be smaller. For instance, suppose my age in base 10 is 32. Written in base 16 it is only 20! However, I cannot choose an arbitrary base when doing this. If my age written in base b contains digits other than 0 to 9, then it will be obvious that I am cheating, which defeats the purpose. In addition, if my age written in base b is too small then it would again be obvious that I am cheating. Given my age y and a lower bound ` on how small I want my age to appear, find the largest base b such that y written in base b contains only decimal digits, and is at least ` when interpreted as a number in base 10.

Input

The input consists of a single line containing two base 10 integers y (10 ≤ y ≤ 1018 – yes, I am very old) and ` (10 ≤ ` ≤ y).

Output

Display the largest base b as described above.

Sample Tests

Sample 1
Sample Input 
 32 20
Sample Output 
16
Sample 2
Sample Input 
 2016 100
Sample Output 
42
Editorial

Editorial

The solution write-up is rendered from the LaTeX source, with equations kept live through MathJax.

Raw TeX

Key Observations

  • Write the structural observations that make the problem tractable.

  • State any useful invariant, monotonicity property, graph interpretation, or combinatorial reformulation.

  • If the constraints matter, explain exactly which part of the solution they enable.

Algorithm

  1. Describe the data structures and the state maintained by the algorithm.

  2. Explain the processing order and why it is sufficient.

  3. Mention corner cases explicitly if they affect the implementation.

Correctness Proof

We prove that the algorithm returns the correct answer.

Lemma 1.

State the first key claim.

Proof.

Provide a concise proof.

Lemma 2.

State the next claim if needed.

Proof.

Provide a concise proof.

Theorem.

The algorithm outputs the correct answer for every valid input.

Proof.

Combine the lemmas and finish the argument.

Complexity Analysis

State the running time and memory usage in terms of the input size.

Implementation Notes

  • Mention any non-obvious implementation detail that is easy to get wrong.

  • Mention numeric limits, indexing conventions, or tie-breaking rules if relevant.

Implementation

Code

C++ solution used for this page.

C++

Clean code view with a raw-file link when you want the original source.

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#include <bits/stdc++.h>
using namespace std;

namespace {

void solve() {
    // Fill in the full solution logic for the problem here.
}

}  // namespace

int main() {
    ios::sync_with_stdio(false);
    cin.tie(nullptr);

    solve();
    return 0;
}
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TeX write-upcompetitive_programming/icpc/2016/E-forever-young/solution.texC++ implementationcompetitive_programming/icpc/2016/E-forever-young/solution.cppStatement textcompetitive_programming/icpc/2016/E-forever-young/statement.txtStatement PDFcompetitive_programming/icpc/2016/E-forever-young/statement.pdfMetadatacompetitive_programming/icpc/2016/E-forever-young/meta.jsonYear packetcompetitive_programming/icpc/2016/contest_problems.pdf