// <copyright file="Euclid.cs" company="Math.NET">
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// Math.NET Numerics, part of the Math.NET Project
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// http://numerics.mathdotnet.com
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// http://github.com/mathnet/mathnet-numerics
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//
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// Copyright (c) 2009-2014 Math.NET
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//
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// Permission is hereby granted, free of charge, to any person
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// obtaining a copy of this software and associated documentation
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// files (the "Software"), to deal in the Software without
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// restriction, including without limitation the rights to use,
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// copy, modify, merge, publish, distribute, sublicense, and/or sell
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// copies of the Software, and to permit persons to whom the
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// Software is furnished to do so, subject to the following
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// conditions:
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//
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// The above copyright notice and this permission notice shall be
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// included in all copies or substantial portions of the Software.
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//
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// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
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// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
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// OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
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// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
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// HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
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// WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
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// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
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// OTHER DEALINGS IN THE SOFTWARE.
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// </copyright>
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using System;
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using System.Collections.Generic;
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using System.Runtime.CompilerServices;
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using BigInteger = System.Numerics.BigInteger;
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namespace IStation.Numerics
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{
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/// <summary>
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/// Integer number theory functions.
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/// </summary>
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public static class Euclid
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{
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/// <summary>
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/// Canonical Modulus. The result has the sign of the divisor.
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/// </summary>
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#if !NET40
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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#endif
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public static double Modulus(double dividend, double divisor)
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{
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return ((dividend%divisor) + divisor)%divisor;
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}
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/// <summary>
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/// Canonical Modulus. The result has the sign of the divisor.
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/// </summary>
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#if !NET40
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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#endif
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public static float Modulus(float dividend, float divisor)
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{
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return ((dividend%divisor) + divisor)%divisor;
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}
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/// <summary>
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/// Canonical Modulus. The result has the sign of the divisor.
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/// </summary>
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#if !NET40
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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#endif
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public static int Modulus(int dividend, int divisor)
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{
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return ((dividend%divisor) + divisor)%divisor;
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}
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/// <summary>
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/// Canonical Modulus. The result has the sign of the divisor.
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/// </summary>
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#if !NET40
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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#endif
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public static long Modulus(long dividend, long divisor)
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{
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return ((dividend%divisor) + divisor)%divisor;
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}
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/// <summary>
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/// Canonical Modulus. The result has the sign of the divisor.
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/// </summary>
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#if !NET40
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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#endif
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public static BigInteger Modulus(BigInteger dividend, BigInteger divisor)
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{
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return ((dividend%divisor) + divisor)%divisor;
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}
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/// <summary>
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/// Remainder (% operator). The result has the sign of the dividend.
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/// </summary>
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#if !NET40
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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#endif
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public static double Remainder(double dividend, double divisor)
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{
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return dividend%divisor;
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}
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/// <summary>
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/// Remainder (% operator). The result has the sign of the dividend.
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/// </summary>
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#if !NET40
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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#endif
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public static float Remainder(float dividend, float divisor)
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{
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return dividend%divisor;
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}
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/// <summary>
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/// Remainder (% operator). The result has the sign of the dividend.
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/// </summary>
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#if !NET40
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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#endif
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public static int Remainder(int dividend, int divisor)
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{
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return dividend%divisor;
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}
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/// <summary>
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/// Remainder (% operator). The result has the sign of the dividend.
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/// </summary>
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#if !NET40
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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#endif
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public static long Remainder(long dividend, long divisor)
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{
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return dividend%divisor;
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}
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/// <summary>
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/// Remainder (% operator). The result has the sign of the dividend.
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/// </summary>
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#if !NET40
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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#endif
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public static BigInteger Remainder(BigInteger dividend, BigInteger divisor)
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{
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return dividend%divisor;
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}
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/// <summary>
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/// Find out whether the provided 32 bit integer is an even number.
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/// </summary>
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/// <param name="number">The number to very whether it's even.</param>
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/// <returns>True if and only if it is an even number.</returns>
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#if !NET40
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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#endif
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public static bool IsEven(this int number)
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{
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return (number & 0x1) == 0x0;
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}
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/// <summary>
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/// Find out whether the provided 64 bit integer is an even number.
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/// </summary>
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/// <param name="number">The number to very whether it's even.</param>
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/// <returns>True if and only if it is an even number.</returns>
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#if !NET40
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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#endif
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public static bool IsEven(this long number)
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{
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return (number & 0x1) == 0x0;
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}
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/// <summary>
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/// Find out whether the provided 32 bit integer is an odd number.
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/// </summary>
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/// <param name="number">The number to very whether it's odd.</param>
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/// <returns>True if and only if it is an odd number.</returns>
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#if !NET40
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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#endif
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public static bool IsOdd(this int number)
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{
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return (number & 0x1) == 0x1;
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}
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/// <summary>
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/// Find out whether the provided 64 bit integer is an odd number.
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/// </summary>
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/// <param name="number">The number to very whether it's odd.</param>
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/// <returns>True if and only if it is an odd number.</returns>
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#if !NET40
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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#endif
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public static bool IsOdd(this long number)
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{
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return (number & 0x1) == 0x1;
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}
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/// <summary>
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/// Find out whether the provided 32 bit integer is a perfect power of two.
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/// </summary>
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/// <param name="number">The number to very whether it's a power of two.</param>
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/// <returns>True if and only if it is a power of two.</returns>
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#if !NET40
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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#endif
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public static bool IsPowerOfTwo(this int number)
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{
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return number > 0 && (number & (number - 1)) == 0x0;
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}
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/// <summary>
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/// Find out whether the provided 64 bit integer is a perfect power of two.
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/// </summary>
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/// <param name="number">The number to very whether it's a power of two.</param>
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/// <returns>True if and only if it is a power of two.</returns>
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#if !NET40
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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#endif
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public static bool IsPowerOfTwo(this long number)
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{
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return number > 0 && (number & (number - 1)) == 0x0;
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}
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/// <summary>
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/// Find out whether the provided 32 bit integer is a perfect square, i.e. a square of an integer.
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/// </summary>
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/// <param name="number">The number to very whether it's a perfect square.</param>
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/// <returns>True if and only if it is a perfect square.</returns>
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public static bool IsPerfectSquare(this int number)
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{
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if (number < 0)
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{
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return false;
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}
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int lastHexDigit = number & 0xF;
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if (lastHexDigit > 9)
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{
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return false; // return immediately in 6 cases out of 16.
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}
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if (lastHexDigit == 0 || lastHexDigit == 1 || lastHexDigit == 4 || lastHexDigit == 9)
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{
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int t = (int)Math.Floor(Math.Sqrt(number) + 0.5);
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return (t * t) == number;
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}
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return false;
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}
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/// <summary>
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/// Find out whether the provided 64 bit integer is a perfect square, i.e. a square of an integer.
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/// </summary>
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/// <param name="number">The number to very whether it's a perfect square.</param>
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/// <returns>True if and only if it is a perfect square.</returns>
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public static bool IsPerfectSquare(this long number)
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{
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if (number < 0)
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{
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return false;
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}
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int lastHexDigit = (int)(number & 0xF);
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if (lastHexDigit > 9)
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{
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return false; // return immediately in 6 cases out of 16.
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}
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if (lastHexDigit == 0 || lastHexDigit == 1 || lastHexDigit == 4 || lastHexDigit == 9)
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{
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long t = (long)Math.Floor(Math.Sqrt(number) + 0.5);
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return (t * t) == number;
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}
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return false;
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}
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/// <summary>
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/// Raises 2 to the provided integer exponent (0 <= exponent < 31).
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/// </summary>
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/// <param name="exponent">The exponent to raise 2 up to.</param>
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/// <returns>2 ^ exponent.</returns>
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/// <exception cref="ArgumentOutOfRangeException"/>
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public static int PowerOfTwo(this int exponent)
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{
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if (exponent < 0 || exponent >= 31)
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{
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throw new ArgumentOutOfRangeException(nameof(exponent));
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}
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return 1 << exponent;
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}
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/// <summary>
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/// Raises 2 to the provided integer exponent (0 <= exponent < 63).
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/// </summary>
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/// <param name="exponent">The exponent to raise 2 up to.</param>
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/// <returns>2 ^ exponent.</returns>
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/// <exception cref="ArgumentOutOfRangeException"/>
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public static long PowerOfTwo(this long exponent)
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{
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if (exponent < 0 || exponent >= 63)
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{
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throw new ArgumentOutOfRangeException(nameof(exponent));
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}
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return ((long)1) << (int)exponent;
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}
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/// <summary>
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/// Evaluate the binary logarithm of an integer number.
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/// </summary>
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/// <remarks>Two-step method using a De Bruijn-like sequence table lookup.</remarks>
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public static int Log2(this int number)
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{
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number |= number >> 1;
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number |= number >> 2;
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number |= number >> 4;
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number |= number >> 8;
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number |= number >> 16;
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return MultiplyDeBruijnBitPosition[(uint)(number * 0x07C4ACDDU) >> 27];
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}
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static readonly int[] MultiplyDeBruijnBitPosition = new int[32]
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{
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0, 9, 1, 10, 13, 21, 2, 29, 11, 14, 16, 18, 22, 25, 3, 30,
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8, 12, 20, 28, 15, 17, 24, 7, 19, 27, 23, 6, 26, 5, 4, 31
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};
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/// <summary>
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/// Find the closest perfect power of two that is larger or equal to the provided
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/// 32 bit integer.
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/// </summary>
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/// <param name="number">The number of which to find the closest upper power of two.</param>
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/// <returns>A power of two.</returns>
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/// <exception cref="ArgumentOutOfRangeException"/>
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public static int CeilingToPowerOfTwo(this int number)
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{
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if (number == Int32.MinValue)
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{
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return 0;
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}
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const int maxPowerOfTwo = 0x40000000;
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if (number > maxPowerOfTwo)
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{
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throw new ArgumentOutOfRangeException(nameof(number));
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}
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number--;
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number |= number >> 1;
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number |= number >> 2;
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number |= number >> 4;
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number |= number >> 8;
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number |= number >> 16;
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return number + 1;
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}
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/// <summary>
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/// Find the closest perfect power of two that is larger or equal to the provided
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/// 64 bit integer.
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/// </summary>
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/// <param name="number">The number of which to find the closest upper power of two.</param>
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/// <returns>A power of two.</returns>
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/// <exception cref="ArgumentOutOfRangeException"/>
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public static long CeilingToPowerOfTwo(this long number)
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{
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if (number == Int64.MinValue)
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{
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return 0;
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}
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const long maxPowerOfTwo = 0x4000000000000000;
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if (number > maxPowerOfTwo)
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{
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throw new ArgumentOutOfRangeException(nameof(number));
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}
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number--;
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number |= number >> 1;
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number |= number >> 2;
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number |= number >> 4;
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number |= number >> 8;
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number |= number >> 16;
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number |= number >> 32;
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return number + 1;
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}
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/// <summary>
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/// Returns the greatest common divisor (<c>gcd</c>) of two integers using Euclid's algorithm.
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/// </summary>
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/// <param name="a">First Integer: a.</param>
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/// <param name="b">Second Integer: b.</param>
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/// <returns>Greatest common divisor <c>gcd</c>(a,b)</returns>
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public static long GreatestCommonDivisor(long a, long b)
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{
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while (b != 0)
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{
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var remainder = a%b;
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a = b;
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b = remainder;
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}
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return Math.Abs(a);
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}
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/// <summary>
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/// Returns the greatest common divisor (<c>gcd</c>) of a set of integers using Euclid's
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/// algorithm.
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/// </summary>
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/// <param name="integers">List of Integers.</param>
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/// <returns>Greatest common divisor <c>gcd</c>(list of integers)</returns>
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public static long GreatestCommonDivisor(IList<long> integers)
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{
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if (null == integers)
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{
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throw new ArgumentNullException(nameof(integers));
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}
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if (integers.Count == 0)
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{
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return 0;
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}
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var gcd = Math.Abs(integers[0]);
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for (var i = 1; (i < integers.Count) && (gcd > 1); i++)
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{
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gcd = GreatestCommonDivisor(gcd, integers[i]);
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}
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return gcd;
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}
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/// <summary>
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/// Returns the greatest common divisor (<c>gcd</c>) of a set of integers using Euclid's algorithm.
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/// </summary>
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/// <param name="integers">List of Integers.</param>
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/// <returns>Greatest common divisor <c>gcd</c>(list of integers)</returns>
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public static long GreatestCommonDivisor(params long[] integers)
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{
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return GreatestCommonDivisor((IList<long>)integers);
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}
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/// <summary>
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/// Computes the extended greatest common divisor, such that a*x + b*y = <c>gcd</c>(a,b).
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/// </summary>
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/// <param name="a">First Integer: a.</param>
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/// <param name="b">Second Integer: b.</param>
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/// <param name="x">Resulting x, such that a*x + b*y = <c>gcd</c>(a,b).</param>
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/// <param name="y">Resulting y, such that a*x + b*y = <c>gcd</c>(a,b)</param>
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/// <returns>Greatest common divisor <c>gcd</c>(a,b)</returns>
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/// <example>
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/// <code>
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/// long x,y,d;
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/// d = Fn.GreatestCommonDivisor(45,18,out x, out y);
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/// -> d == 9 && x == 1 && y == -2
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/// </code>
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/// 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.
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/// </example>
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public static long ExtendedGreatestCommonDivisor(long a, long b, out long x, out long y)
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{
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long mp = 1, np = 0, m = 0, n = 1;
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while (b != 0)
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{
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long rem;
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#if NETSTANDARD1_3
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rem = a % b;
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var quot = a / b;
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#else
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long quot = Math.DivRem(a, b, out rem);
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#endif
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a = b;
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b = rem;
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var tmp = m;
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m = mp - (quot*m);
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mp = tmp;
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tmp = n;
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n = np - (quot*n);
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np = tmp;
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}
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if (a >= 0)
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{
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x = mp;
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y = np;
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return a;
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}
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x = -mp;
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y = -np;
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return -a;
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}
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/// <summary>
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/// Returns the least common multiple (<c>lcm</c>) of two integers using Euclid's algorithm.
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/// </summary>
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/// <param name="a">First Integer: a.</param>
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/// <param name="b">Second Integer: b.</param>
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/// <returns>Least common multiple <c>lcm</c>(a,b)</returns>
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public static long LeastCommonMultiple(long a, long b)
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{
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if ((a == 0) || (b == 0))
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{
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return 0;
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}
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return Math.Abs((a/GreatestCommonDivisor(a, b))*b);
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}
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/// <summary>
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/// Returns the least common multiple (<c>lcm</c>) of a set of integers using Euclid's algorithm.
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/// </summary>
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/// <param name="integers">List of Integers.</param>
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/// <returns>Least common multiple <c>lcm</c>(list of integers)</returns>
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public static long LeastCommonMultiple(IList<long> integers)
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{
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if (null == integers)
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{
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throw new ArgumentNullException(nameof(integers));
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}
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if (integers.Count == 0)
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{
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return 1;
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}
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var lcm = Math.Abs(integers[0]);
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for (var i = 1; i < integers.Count; i++)
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{
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lcm = LeastCommonMultiple(lcm, integers[i]);
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}
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return lcm;
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}
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/// <summary>
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/// Returns the least common multiple (<c>lcm</c>) of a set of integers using Euclid's algorithm.
|
/// </summary>
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/// <param name="integers">List of Integers.</param>
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/// <returns>Least common multiple <c>lcm</c>(list of integers)</returns>
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public static long LeastCommonMultiple(params long[] integers)
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{
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return LeastCommonMultiple((IList<long>)integers);
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}
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/// <summary>
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/// Returns the greatest common divisor (<c>gcd</c>) of two big integers.
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/// </summary>
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/// <param name="a">First Integer: a.</param>
|
/// <param name="b">Second Integer: b.</param>
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/// <returns>Greatest common divisor <c>gcd</c>(a,b)</returns>
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public static BigInteger GreatestCommonDivisor(BigInteger a, BigInteger b)
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{
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return BigInteger.GreatestCommonDivisor(a, b);
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}
|
|
/// <summary>
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/// Returns the greatest common divisor (<c>gcd</c>) of a set of big integers.
|
/// </summary>
|
/// <param name="integers">List of Integers.</param>
|
/// <returns>Greatest common divisor <c>gcd</c>(list of integers)</returns>
|
public static BigInteger GreatestCommonDivisor(IList<BigInteger> integers)
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{
|
if (null == integers)
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{
|
throw new ArgumentNullException(nameof(integers));
|
}
|
|
if (integers.Count == 0)
|
{
|
return 0;
|
}
|
|
var gcd = BigInteger.Abs(integers[0]);
|
|
for (int i = 1; (i < integers.Count) && (gcd > BigInteger.One); i++)
|
{
|
gcd = GreatestCommonDivisor(gcd, integers[i]);
|
}
|
|
return gcd;
|
}
|
|
/// <summary>
|
/// Returns the greatest common divisor (<c>gcd</c>) of a set of big integers.
|
/// </summary>
|
/// <param name="integers">List of Integers.</param>
|
/// <returns>Greatest common divisor <c>gcd</c>(list of integers)</returns>
|
public static BigInteger GreatestCommonDivisor(params BigInteger[] integers)
|
{
|
return GreatestCommonDivisor((IList<BigInteger>)integers);
|
}
|
|
/// <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;
|
}
|
|
/// <summary>
|
/// Returns the least common multiple (<c>lcm</c>) of two big integers.
|
/// </summary>
|
/// <param name="a">First Integer: a.</param>
|
/// <param name="b">Second Integer: b.</param>
|
/// <returns>Least common multiple <c>lcm</c>(a,b)</returns>
|
public static BigInteger LeastCommonMultiple(BigInteger a, BigInteger b)
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{
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if (a.IsZero || b.IsZero)
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{
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return BigInteger.Zero;
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}
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return BigInteger.Abs((a/BigInteger.GreatestCommonDivisor(a, b))*b);
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}
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/// <summary>
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/// Returns the least common multiple (<c>lcm</c>) of a set of big integers.
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/// </summary>
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/// <param name="integers">List of Integers.</param>
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/// <returns>Least common multiple <c>lcm</c>(list of integers)</returns>
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public static BigInteger LeastCommonMultiple(IList<BigInteger> integers)
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{
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if (null == integers)
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{
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throw new ArgumentNullException(nameof(integers));
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}
|
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if (integers.Count == 0)
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{
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return 1;
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}
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var lcm = BigInteger.Abs(integers[0]);
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for (int i = 1; i < integers.Count; i++)
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{
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lcm = LeastCommonMultiple(lcm, integers[i]);
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}
|
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return lcm;
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}
|
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/// <summary>
|
/// Returns the least common multiple (<c>lcm</c>) of a set of big integers.
|
/// </summary>
|
/// <param name="integers">List of Integers.</param>
|
/// <returns>Least common multiple <c>lcm</c>(list of integers)</returns>
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public static BigInteger LeastCommonMultiple(params BigInteger[] integers)
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{
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return LeastCommonMultiple((IList<BigInteger>)integers);
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}
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}
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}
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