/// <summary>
/// Gets the least common multiple of two elements in a <see cref="IEuclideanDomain{T, TFirst, TSecond}"/> via Euclid's algorithm. The LCDM is defined as:
/// <code>
/// r.Mult(r.Gcd(x,y), r.Lcm(x,y)) == r.Mult(x,y)
/// </code>
/// </summary>
/// <typeparam name="T">The type of the carrier set.</typeparam>
/// <typeparam name="TFirst">The type of the first grouplike operation.</typeparam>
/// <typeparam name="TSecond">The type of the second groupike operation.</typeparam>
/// <param name="r">The ringlike structure.</param>
/// <param name="x">The first element.</param>
/// <param name="y">The second element.</param>
/// <exception cref="DivideByZeroException">If <paramref name="x"/> or <paramref name="y"/> is zero.</exception>
public static T Lcm <T, TFirst, TSecond>(this IEuclideanDomain <T, TFirst, TSecond> r, T x, T y)
where TFirst : ICommutativeGroup <T>
where TSecond : IMonoid <T>, ICommutative <T>
{
if (r.IsZero(x) || r.IsZero(y))
{
throw new DivideByZeroException();
}
var gcd = r.Gcd(x, y);
var numerator = r.Mult(x, y);
var ret = r.EuclideanDivide(numerator, gcd).quotient;
return(ret);
}
/// <summary>
/// Gets the greatest common divisor of two elements in a <see cref="IEuclideanDomain{T, TFirst, TSecond}"/> via Euclid's algorithm. The GCD of two elements <c>X</c> and <c>Y</c> is the unique minimal principal ideal.
/// </summary>
/// <typeparam name="T">The type of the carrier set.</typeparam>
/// <typeparam name="TFirst">The type of the first grouplike operation.</typeparam>
/// <typeparam name="TSecond">The type of the second groupike operation.</typeparam>
/// <param name="r">The ringlike structure.</param>
/// <param name="x">The first element.</param>
/// <param name="y">The second element.</param>
public static T Gcd <T, TFirst, TSecond>(this IEuclideanDomain <T, TFirst, TSecond> r, T x, T y)
where TFirst : ICommutativeGroup <T>
where TSecond : IMonoid <T>, ICommutative <T>
{
if (r.IsZero(x) || r.IsZero(y))
{
return(r.Zero <T, TFirst>());
}
while (!r.IsZero(y))
{
var t = y;
y = r.EuclideanDivide(x, y).remainder;
x = t;
}
return(x);
}
/// <summary>
/// Performs the modulus-operation on two elements, returning the remainder of <see cref="IEuclideanDomain{T, TFirst, TSecond}.EuclideanDivide(T, T)"/>.
/// </summary>
/// <typeparam name="T">The type of the carrier set.</typeparam>
/// <typeparam name="TFirst">The type of the first grouplike operation.</typeparam>
/// <typeparam name="TSecond">The type of the second groupike operation.</typeparam>
/// <param name="r">The ringlike structure.</param>
/// <param name="x">The first element.</param>
/// <param name="y">The second element.</param>
public static T Mod <T, TFirst, TSecond>(this IEuclideanDomain <T, TFirst, TSecond> r, T x, T y)
where TFirst : ICommutativeGroup <T>
where TSecond : IMonoid <T>, ICommutative <T>
=> r.EuclideanDivide(x, y).remainder;