/// <summary>
/// Computes the extended greatest common divisor, such that a*x + b*y = <c>gcd</c>(a,b).
/// </summary>
/// <param name="a">First Integer: a.</param>
/// <param name="b">Second Integer: b.</param>
/// <param name="x">Resulting x, such that a*x + b*y = <c>gcd</c>(a,b).</param>
/// <param name="y">Resulting y, such that a*x + b*y = <c>gcd</c>(a,b)</param>
/// <returns>Greatest common divisor <c>gcd</c>(a,b)</returns>
/// <example>
/// <code>
/// long x,y,d;
/// d = Fn.GreatestCommonDivisor(45,18,out x, out y);
/// -> d == 9 && x == 1 && y == -2
/// </code>
/// The <c>gcd</c> of 45 and 18 is 9: 18 = 2*9, 45 = 5*9. 9 = 1*45 -2*18, therefore x=1 and y=-2.
/// </example>
public static BigInteger ExtendedGreatestCommonDivisor(BigInteger a, BigInteger b, out BigInteger x, out BigInteger y)
{
BigInteger mp = BigInteger.One, np = BigInteger.Zero, m = BigInteger.Zero, n = BigInteger.One;
while (!b.IsZero)
{
BigInteger rem;
BigInteger quot = BigInteger.DivRem(a, b, out rem);
a = b;
b = rem;
BigInteger tmp = m;
m = mp - (quot * m);
mp = tmp;
tmp = n;
n = np - (quot * n);
np = tmp;
}
if (a >= BigInteger.Zero)
{
x = mp;
y = np;
return(a);
}
x = -mp;
y = -np;
return(-a);
}
public static BigInteger DivRem(BigInteger x, BigInteger y, out BigInteger remainder)
{
BigInt rem;
BigInt result = BigInt.DivRem(x.Value, y.Value, out rem);
remainder = new BigInteger(rem);
return(new BigInteger(result));
}