[Projecteuler.net] 5; 6; 9

5
 

 
Euclidean Least Common Divisor
 

 
F # implementation:
 

 
/// Euclidean Algorithm// Http://en.wikipedia.org/wiki/Euclidean_algorithm/// Calculate the maximum common divisor of a B/// Use the moving phase Subtraction Method/// The original ooriAlgorithmYes:/// Set C to the maximum public approx./// A = ac B = BC/// Gcd (a, B) => gcd (A-B, B) [A> B] and gcd (a-B) C, B) constantly use big minus small ones. Until a-B = 1, C is returned.
 

 
Let RECGCD a B =MatchAWith| XWhenX = 0i->B | _->Gcd (B % A)
 

It's easy to have the least common approx.
 

 
LetLCM a B = a * B/(gcd a B) the fifth question is to find the minimum public multiple of 1 to 20.
 

 
So long as this can be:
 

 
Let RECLcms xs =MatchXSWith| A: B: T->Lcms (lcm a B: t) | A: []->A
 

 
Lcms [1 .. 20]
 

 
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6

Note two formulas first: (1 + 2 + 3... + n) ^ 2 = (1 + n) * n/2) ^ 2 (1 ^ 2 + 2 ^ 2 + 3 ^ 2 +... + N ^ 2) = 1/6 * n (n + 1) (2n + 1) ========================================================== ========================================================== ====================
 

 

9

 /// A Pythagorean triplet check  Let Ispt a B c = A * A + B * B = C * C Let Sumeq1000 a B c = A + B + C = 1000 Let Help predicate = Let  Mutable Ret = [] For AIn 1. 998 Do  For B In (A + 1) .. 999 Do  For C In (B + 1) .. 1000 Do  If Ispt a B c & predicate a B c Then RET <-(A, B, C): Ret RET Let Findpt = help sumeq1000

 // Prie solution: http://projecteuler.net/thread=9  /// Without programming:  ///  /// A = 2mn; B = m ^ 2-N ^ 2; C = m ^ 2 + N ^ 2;  /// A + B + C = 1000;  ///  /// 2mn + (M ^ 2-N ^ 2) + (M ^ 2 + N ^ 2) = 1000;  /// 2mn + 2 m ^ 2 = 1000;  /// 2 m (m + n) = 1000;  /// M (m + n) = 500; ///  /// M> N;  ///  /// M = 20; n = 5;  ///  /// A = 200; B = 375; C = 425;  /// Pythagorean triplet K will be:  /// K {M ^ 2-N ^ 2, 2 m n, m ^ 2 + N ^ 2}  /// X = A + B + C  Let Ptpire x = Let  Mutable Ret = [] For M In 1. X/2 Do   For N In M-1 Do  Let A = m * m-N * n In  Let B = 2 * m * n In  Let C = m * m + n * n In  If A + B + C = x & A * A + B * B = C * C Then RET <-(A, B, C): Ret RET