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// Copyright © 2026 Mikhail Hogrefe
//
// This file is part of Malachite.
//
// Malachite is free software: you can redistribute it and/or modify it under the terms of the GNU
// Lesser General Public License (LGPL) as published by the Free Software Foundation; either version
// 3 of the License, or (at your option) any later version. See <https://www.gnu.org/licenses/>.
use crate::natural::Natural;
use malachite_base::num::arithmetic::traits::{DivExact, DivExactAssign, Gcd, Lcm, LcmAssign};
use malachite_base::num::basic::traits::Zero;
impl Lcm<Self> for Natural {
type Output = Self;
/// Computes the LCM (least common multiple) of two [`Natural`]s, taking both by value.
///
/// $$
/// f(x, y) = \operatorname{lcm}(x, y).
/// $$
///
/// # Worst-case complexity
/// $T(n) = O(n (\log n)^2 \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `max(self.significant_bits(),
/// other.significant_bits())`.
///
/// # Examples
/// ```
/// use malachite_base::num::arithmetic::traits::Lcm;
/// use malachite_nz::natural::Natural;
///
/// assert_eq!(Natural::from(3u32).lcm(Natural::from(5u32)), 15);
/// assert_eq!(Natural::from(12u32).lcm(Natural::from(90u32)), 180);
/// ```
fn lcm(mut self, other: Self) -> Self {
self.lcm_assign(other);
self
}
}
impl<'a> Lcm<&'a Self> for Natural {
type Output = Self;
/// Computes the LCM (least common multiple) of two [`Natural`]s, taking the first by value and
/// the second by reference.
///
/// $$
/// f(x, y) = \operatorname{lcm}(x, y).
/// $$
///
/// # Worst-case complexity
/// $T(n) = O(n (\log n)^2 \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `max(self.significant_bits(),
/// other.significant_bits())`.
///
/// # Examples
/// ```
/// use malachite_base::num::arithmetic::traits::Lcm;
/// use malachite_nz::natural::Natural;
///
/// assert_eq!(Natural::from(3u32).lcm(&Natural::from(5u32)), 15);
/// assert_eq!(Natural::from(12u32).lcm(&Natural::from(90u32)), 180);
/// ```
#[inline]
fn lcm(mut self, other: &'a Self) -> Self {
self.lcm_assign(other);
self
}
}
impl Lcm<Natural> for &Natural {
type Output = Natural;
/// Computes the LCM (least common multiple) of two [`Natural`]s, taking the first by reference
/// and the second by value.
///
/// $$
/// f(x, y) = \operatorname{lcm}(x, y).
/// $$
///
/// # Worst-case complexity
/// $T(n) = O(n (\log n)^2 \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `max(self.significant_bits(),
/// other.significant_bits())`.
///
/// # Examples
/// ```
/// use malachite_base::num::arithmetic::traits::Lcm;
/// use malachite_nz::natural::Natural;
///
/// assert_eq!((&Natural::from(3u32)).lcm(Natural::from(5u32)), 15);
/// assert_eq!((&Natural::from(12u32)).lcm(Natural::from(90u32)), 180);
/// ```
#[inline]
fn lcm(self, mut other: Natural) -> Natural {
other.lcm_assign(self);
other
}
}
impl Lcm<&Natural> for &Natural {
type Output = Natural;
/// Computes the LCM (least common multiple) of two [`Natural`]s, taking both by reference.
///
/// $$
/// f(x, y) = \operatorname{lcm}(x, y).
/// $$
///
/// # Worst-case complexity
/// $T(n) = O(n (\log n)^2 \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `max(self.significant_bits(),
/// other.significant_bits())`.
///
/// # Examples
/// ```
/// use malachite_base::num::arithmetic::traits::Lcm;
/// use malachite_nz::natural::Natural;
///
/// assert_eq!((&Natural::from(3u32)).lcm(&Natural::from(5u32)), 15);
/// assert_eq!((&Natural::from(12u32)).lcm(&Natural::from(90u32)), 180);
/// ```
#[inline]
fn lcm(self, other: &Natural) -> Natural {
if *self == 0 || *other == 0 {
return Natural::ZERO;
}
let gcd = self.gcd(other);
// Division is slower than multiplication, so we choose the arguments to div_exact to be as
// small as possible. This also allows the special case of lcm(x, y) when x is a multiple of
// y to be quickly reduced to x.
if self >= other {
self * other.div_exact(gcd)
} else {
other * self.div_exact(gcd)
}
}
}
impl LcmAssign<Self> for Natural {
/// Replaces a [`Natural`] by its LCM (least common multiple) with another [`Natural`], taking
/// the [`Natural`] on the right-hand side by value.
///
/// $$
/// x \gets \operatorname{lcm}(x, y).
/// $$
///
/// # Worst-case complexity
/// $T(n) = O(n (\log n)^2 \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `max(self.significant_bits(),
/// other.significant_bits())`.
///
/// # Examples
/// ```
/// use malachite_base::num::arithmetic::traits::LcmAssign;
/// use malachite_nz::natural::Natural;
///
/// let mut x = Natural::from(3u32);
/// x.lcm_assign(Natural::from(5u32));
/// assert_eq!(x, 15);
///
/// let mut x = Natural::from(12u32);
/// x.lcm_assign(Natural::from(90u32));
/// assert_eq!(x, 180);
/// ```
#[inline]
fn lcm_assign(&mut self, mut other: Self) {
if *self == 0 {
return;
} else if other == 0 {
*self = Self::ZERO;
return;
}
let gcd = (&*self).gcd(&other);
if *self >= other {
other.div_exact_assign(gcd);
} else {
self.div_exact_assign(gcd);
}
*self *= other;
}
}
impl<'a> LcmAssign<&'a Self> for Natural {
/// Replaces a [`Natural`] by its LCM (least common multiple) with another [`Natural`], taking
/// the [`Natural`] on the right-hand side by reference.
///
/// $$
/// x \gets \operatorname{lcm}(x, y).
/// $$
///
/// # Worst-case complexity
/// $T(n) = O(n (\log n)^2 \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `max(self.significant_bits(),
/// other.significant_bits())`.
///
/// # Examples
/// ```
/// use malachite_base::num::arithmetic::traits::LcmAssign;
/// use malachite_nz::natural::Natural;
///
/// let mut x = Natural::from(3u32);
/// x.lcm_assign(&Natural::from(5u32));
/// assert_eq!(x, 15);
///
/// let mut x = Natural::from(12u32);
/// x.lcm_assign(&Natural::from(90u32));
/// assert_eq!(x, 180);
/// ```
#[inline]
fn lcm_assign(&mut self, other: &'a Self) {
if *self == 0 {
return;
} else if *other == 0 {
*self = Self::ZERO;
return;
}
self.div_exact_assign((&*self).gcd(other));
*self *= other;
}
}