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 Rust (1.42.0) (judge verdict: Accepted)

#![allow(unused_imports, unused_macros)]

use kyoproio::*;
use std::{
    collections::*,
    io::{self, prelude::*},
    iter,
    mem::{replace, swap},
};

fn main() -> io::Result<()> {
    std::thread::Builder::new()
        .stack_size(64 * 1024 * 1024)
        .spawn(|| {
            let stdin = io::stdin();
            let stdout = io::stdout();
            run(KInput::new(stdin.lock()), io::BufWriter::new(stdout.lock()))
        })?
        .join()
        .unwrap()
}

fn run<I: Input, O: Write>(mut kin: I, mut out: O) -> io::Result<()> {
    macro_rules! output { ($($args:expr),+) => { write!(&mut out, $($args),+)?; }; }
    macro_rules! outputln {
        ($($args:expr),+) => { output!($($args),+); outputln!(); };
        () => { output!("\n"); if cfg!(debug_assertions) { out.flush()?; } }
    }

    let n: i64 = kin.input();
    if n == 1 {
        outputln!("1");
        return Ok(());
    }
    let ps = primes(((n as f64).sqrt() + 1.) as usize);
    
    let mut fs = Vec::new();
    let mut x = 2 * n;
    for p in ps {
        let p = p as i64;
        if x % p == 0 {
            let mut d = 1;
            while x % p == 0 {
                x /= p;
                d *= p;
            }
            fs.push(d);
        }
    }
    if x > 1 {
        fs.push(x);
    }

    let mut ans = n - 1;
    for s in 0..1 << fs.len() {
        let mut x = 1;
        for i in 0..fs.len() {
            if s & (1 << i) != 0 {
                x *= fs[i];
            }
        }
        let y = 2 * n / x;
        let (g, a, b) = extgcd(x, y);
        if a < 0 && b > 0 {
            ans = ans.min(-a * x);
        }
    }
    outputln!("{}", ans);

    Ok(())
}

pub fn primes(n: usize) -> Vec<usize> {
    // 1, 7, 11, 13, 17, 19, 23, 29
    const SKIP: [u8; 8] = [6, 4, 2, 4, 2, 4, 6, 2];
    const XTOI: [u8; 15] = [
        0, 0, 0, 1, 0, 2, 3, 0, 4, 5, 0, 6, 0, 0, 7,
    ];
    let mut sieve = vec![0u8; n / 30 + 1];
    let mut ps = vec![2, 3, 5];
    if n <= 4 {
        ps.truncate([0, 0, 1, 2, 2][n]);
        return ps;
    }
    let mut x = 7;
    let mut i = 1;
    while x <= n {
        if sieve[i / 8] & 1 << i % 8 == 0 {
            ps.push(x);
            let mut j = i;
            let mut y = x * x;
            while y <= n {
                sieve[y / 30] |= 1 << XTOI[y / 2 % 15];
                y += x * SKIP[j % 8] as usize;
                j += 1;
            }
        }
        x += SKIP[i % 8] as usize;
        i += 1;
    }
    ps
}
pub fn extgcd(a: i64, b: i64) -> (i64, i64, i64) {
    if b != 0 {
        let (g, y, x) = extgcd(b, a.rem_euclid(b));
        (g, x, y - a / b * x)
    } else {
        (a, 1, 0)
    }
}

// -----------------------------------------------------------------------------
pub mod kyoproio {
    use std::io::prelude::*;
    pub trait Input {
        fn str(&mut self) -> &str;
        fn input<T: InputParse>(&mut self) -> T {
            T::input(self)
        }
        fn iter<T: InputParse>(&mut self) -> Iter<T, Self> {
            Iter(self, std::marker::PhantomData)
        }
        fn seq<T: InputParse, B: std::iter::FromIterator<T>>(&mut self, n: usize) -> B {
            self.iter().take(n).collect()
        }
    }
    pub struct KInput<R> {
        src: R,
        buf: String,
        pos: usize,
    }
    impl<R: BufRead> KInput<R> {
        pub fn new(src: R) -> Self {
            Self {
                src,
                buf: String::with_capacity(1024),
                pos: 0,
            }
        }
    }
    impl<R: BufRead> Input for KInput<R> {
        fn str(&mut self) -> &str {
            loop {
                if self.pos >= self.buf.len() {
                    self.pos = 0;
                    self.buf.clear();
                    if self.src.read_line(&mut self.buf).expect("io error") == 0 {
                        return &self.buf;
                    }
                }
                let range = self.pos
                    ..self.buf[self.pos..]
                        .find(|c: char| c.is_ascii_whitespace())
                        .map(|i| i + self.pos)
                        .unwrap_or_else(|| self.buf.len());
                self.pos = range.end + 1;
                if range.end > range.start {
                    return &self.buf[range];
                }
            }
        }
    }
    pub struct Iter<'a, T, I: ?Sized>(&'a mut I, std::marker::PhantomData<*const T>);
    impl<'a, T: InputParse, I: Input + ?Sized> Iterator for Iter<'a, T, I> {
        type Item = T;
        fn next(&mut self) -> Option<T> {
            Some(self.0.input())
        }
    }
    pub trait InputParse: Sized {
        fn input<I: Input + ?Sized>(src: &mut I) -> Self;
    }
    impl InputParse for Vec<u8> {
        fn input<I: Input + ?Sized>(src: &mut I) -> Self {
            src.str().as_bytes().to_owned()
        }
    }
    macro_rules! from_str_impl {
        { $($T:ty)* } => {
            $(impl InputParse for $T {
                fn input<I: Input + ?Sized>(src: &mut I) -> Self {
                    src.str().parse::<$T>().expect("parse error")
                }
            })*
        }
    }
    from_str_impl! {
        String char bool f32 f64 isize i8 i16 i32 i64 i128 usize u8 u16 u32 u64 u128
    }
    macro_rules! tuple_impl {
        ($H:ident $($T:ident)*) => {
            impl<$H: InputParse, $($T: InputParse),*> InputParse for ($H, $($T),*) {
                fn input<I: Input + ?Sized>(src: &mut I) -> Self {
                    ($H::input(src), $($T::input(src)),*)
                }
            }
            tuple_impl!($($T)*);
        };
        () => {}
    }
    tuple_impl!(A B C D E F G);
    #[macro_export]
    macro_rules! kdbg {
        ($($v:expr),*) => {
            if cfg!(debug_assertions) { dbg!($($v),*) } else { ($($v),*) }
        }
    }
}