Problem:
Score : 600 points

Problem Statement

Given is an integer N.
Find the minimum possible positive integer k such that (1+2+\cdots+k) is a multiple of N.
It can be proved that such a positive integer k always exists.

Constraints

1 \leq N \leq 10^{15}

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N

Output

Print the answer in a line.

Sample Input 1

11

Sample Output 1

10

1+2+\cdots+10=55 holds and 55 is indeed a multple of N=11.
There are no positive integers k \leq 9 that satisfy the condition, so the answer is k = 10.

Sample Input 2

20200920

Sample Output 2

1100144

Solution in Java (OpenJDK 11.0.6) (judge verdict: Wrong Answer)

import java.util.*;

@SuppressWarnings("unused")
public class Main {

  private static void solve() {
    long n = nl();
    int[] primes = sieveEratosthenes(100000000);
    n *= 2;
    long[] divs = enumDivisors(n, primes);

    // k(k+1) = p * d * e
    long ret = Long.MAX_VALUE;
    for (long d : divs) {
      long e = n / d;
      long[] g = exgcd(d, e);

      if (g[0] != 1) {
        continue;
      }
      long x = g[1];
      long y = g[2];
      // dx + ey = 1
      // x = x - ek
      // y = y + dk
      if (x < 0) {
        // ey = 1 + d(-x) = k
        long k = d * (-x);
        ret = Math.min(k, ret);
      }

      if (y < 0) {
        // dx = 1 + e(-y) = k
        long k = e * (-y);
        ret = Math.min(k, ret);
      }
    }
    System.out.println(ret);
  }

  public static long[] exgcd(long a, long b) {
    if (a == 0 || b == 0)
      return null;
    int as = Long.signum(a);
    int bs = Long.signum(b);
    a = Math.abs(a);
    b = Math.abs(b);
    long p = 1, q = 0, r = 0, s = 1;
    while (b > 0) {
      long c = a / b;
      long d;
      d = a;
      a = b;
      b = d % b;
      d = p;
      p = q;
      q = d - c * q;
      d = r;
      r = s;
      s = d - c * s;
    }
    return new long[] { a, p * as, r * bs };
  }

  public static int[] sieveEratosthenes(int n) {
    if (n <= 32) {
      int[] primes = { 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 };
      for (int i = 0; i < primes.length; i++) {
        if (n < primes[i]) {
          return Arrays.copyOf(primes, i);
        }
      }
      return primes;
    }

    int u = n + 32;
    double lu = Math.log(u);
    int[] ret = new int[(int) (u / lu + u / lu / lu * 1.5)];
    ret[0] = 2;
    int pos = 1;

    int[] isnp = new int[(n + 1) / 32 / 2 + 1];
    int sup = (n + 1) / 32 / 2 + 1;

    int[] tprimes = { 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 };
    for (int tp : tprimes) {
      ret[pos++] = tp;
      int[] ptn = new int[tp];
      for (int i = (tp - 3) / 2; i < tp << 5; i += tp)
        ptn[i >> 5] |= 1 << i;
      for (int j = 0; j < sup; j += tp) {
        for (int i = 0; i < tp && i + j < sup; i++) {
          isnp[j + i] |= ptn[i];
        }
      }
    }

    // 3,5,7
    // 2x+3=n
    int[] magic = { 0, 1, 23, 2, 29, 24, 19, 3, 30, 27, 25, 11, 20, 8, 4, 13, 31, 22, 28, 18, 26, 10, 7, 12, 21, 17, 9,
        6, 16, 5, 15, 14 };
    int h = n / 2;
    for (int i = 0; i < sup; i++) {
      for (int j = ~isnp[i]; j != 0; j &= j - 1) {
        int pp = i << 5 | magic[(j & -j) * 0x076be629 >>> 27];
        int p = 2 * pp + 3;
        if (p > n)
          break;
        ret[pos++] = p;
        if ((long) p * p > n)
          continue;
        for (int q = (p * p - 3) / 2; q <= h; q += p)
          isnp[q >> 5] |= 1 << q;
      }
    }

    return Arrays.copyOf(ret, pos);
  }

  public static long[] enumDivisors(long n, int[] primes) {
    int m = 1;
    long[] divs = { 1L };
    int t = 1;
    long u = n;
    for (int p : primes) {
      if ((long) p * p > u)
        break;
      int e = 0;
      while (u % p == 0) {
        u /= p;
        e++;
      }
      if (e > 0) {
        divs = Arrays.copyOf(divs, t * (e + 1));
        for (int q = t - 1; q >= 0; q--) {
          long b = divs[q];
          for (int k = 0; k < e; k++) {
            b *= p;
            // if((long)b*b <= n)divs[t++] = b;
            divs[t++] = b;
          }
        }
        m *= e + 1;
      }
    }
    if (u > 1) {
      long p = u;
      int e = 1;
      divs = Arrays.copyOf(divs, t * (e + 1));
      for (int q = t - 1; q >= 0; q--) {
        long b = divs[q];
        for (int k = 0; k < e; k++) {
          b *= p;
          // if((long)b*b <= n)divs[t++] = b;
          divs[t++] = b;
        }
      }
      m *= e + 1;
    }

    return Arrays.copyOf(divs, t);
  }

  public static void main(String[] args) {
    new Thread(null, new Runnable() {
      @Override
      public void run() {
        long start = System.currentTimeMillis();
        String debug = args.length > 0 ? args[0] : null;
        if (debug != null) {
          try {
            is = java.nio.file.Files.newInputStream(java.nio.file.Paths.get(debug));
          } catch (Exception e) {
            throw new RuntimeException(e);
          }
        }
        reader = new java.io.BufferedReader(new java.io.InputStreamReader(is), 32768);
        solve();
        out.flush();
        tr((System.currentTimeMillis() - start) + "ms");
      }
    }, "", 64000000).start();
  }

  private static java.io.InputStream is = System.in;
  private static java.io.PrintWriter out = new java.io.PrintWriter(System.out);
  private static java.util.StringTokenizer tokenizer = null;
  private static java.io.BufferedReader reader;

  public static String next() {
    while (tokenizer == null || !tokenizer.hasMoreTokens()) {
      try {
        tokenizer = new java.util.StringTokenizer(reader.readLine());
      } catch (Exception e) {
        throw new RuntimeException(e);
      }
    }
    return tokenizer.nextToken();
  }

  private static double nd() {
    return Double.parseDouble(next());
  }

  private static long nl() {
    return Long.parseLong(next());
  }

  private static int[] na(int n) {
    int[] a = new int[n];
    for (int i = 0; i < n; i++)
      a[i] = ni();
    return a;
  }

  private static char[] ns() {
    return next().toCharArray();
  }

  private static long[] nal(int n) {
    long[] a = new long[n];
    for (int i = 0; i < n; i++)
      a[i] = nl();
    return a;
  }

  private static int[][] ntable(int n, int m) {
    int[][] table = new int[n][m];
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < m; j++) {
        table[i][j] = ni();
      }
    }
    return table;
  }

  private static int[][] nlist(int n, int m) {
    int[][] table = new int[m][n];
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < m; j++) {
        table[j][i] = ni();
      }
    }
    return table;
  }

  private static int ni() {
    return Integer.parseInt(next());
  }

  private static void tr(Object... o) {
    if (is != System.in)
      System.out.println(java.util.Arrays.deepToString(o));
  }
}