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ICPC 2013
ICPC 2013

D. Factors

2013 World Finals St. Petersburg HOSTED BY ITMO Problem D Factors Time Limit: 2 seconds The fundamental theorem of arithmetic states that every integer greater than 1 can be uniquely repre- sented as a product of one or more primes. While unique, several...

Updated May 21, 2026
Track ICPC
Year 2013
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 2 seconds

The fundamental theorem of arithmetic states that every integer greater than 1 can be uniquely repre- sented as a product of one or more primes. While unique, several arrangements of the prime factors may be possible. For example:

10 = 2 · 5                            20 = 2 · 2 · 5
   =5·2                                   =2·5·2
                                          =5·2·2

Let f (k) be the number of different arrangements of the prime factors of k. So f (10) = 2 and f (20) = 3. Given a positive number n, there always exists at least one number k such that f (k) = n. We want to know the smallest such k.

Input

The input consists of at most 1 000 test cases, each on a separate line. Each test case is a positive integer n < 263 .

Output

For each test case, display its number n and the smallest number k > 1 such that f (k) = n. The numbers in the input are chosen such that k < 263 .

Sample Tests

Sample 1
Sample Input 
 1
 2
 3
 105
Sample Output 
1 2
2 6
3 12
105 720
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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