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Mathematics 1

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4. 1. Permutation

func perm(_ array:[Int],_ depth:Int,_ n:Int,_ k:Int) {
    var array = array
    
    if depth == k {
        print(array)
    } else {
        for i in depth ..< n {
            //SWAP 1
            (array[i],array[depth]) = (array[depth],array[i])
            
            //Resursion
            perm(array, depth + 1, n, k)
            
            //SWAP 2
            (array[i],array[depth]) = (array[depth],array[i])
        }
    }
}

4. 2. Combination

Combinations: Unordered lists

func combination(_ array:[Int],_ index:Int,_ n:Int,_ r:Int,_ target:Int) {
    var array = array
    
    if (r == 0) {
        print(array[0..<index])
    } else if target != n {
        array[index] = target
        combination(array, index + 1, n, r - 1, target + 1)
        combination(array, index, n, r, target + 1)
    }
}
combination([0,1,2,3,4], 0, 5, 3, 0)

4. 3. Prime Number

• If num is divided by i, it is not a prime number.
• If it divided, escape the loop
• divide the prime number smaller than the input number

func getPrime(_ num:Int) {   
    var prime:[Int] = [2]
    
    for i in 2...num {
        for j in 0 ..< prime.count {
            // Not a prime number
            if i % prime[j] == 0 {
                break
            } else if j + 1 == prime.count { // it is prime number
                prime.append(i)
            }
        }
    }
    print(prime)
}
getPrime(10)

4. 4. Eratostheneen seula

Let’s say you want to find all prime number to 120.

After putting them all in the array from 2 to 120, check everything that is not a prime number.
Check all multiples of 2 except 2.
Check all multiples of 3 except 3.
4 is skip because it was checked before.
Check for multiples of all 5 except 5. … …
There are prime numbers of unchecked numbers.

When checking, use the loop only as many times as you want to check without having to check all the numbers. And does not find a multiple of the number already checked to zero. Because the multiples of the checked number are already checked.

func getPrimeNumberEratostheneen(_ num:Int) -> [Int] {
    var array = [Int]()
    var result = [Int]()
    
    //init the array
    for i in 0...num {
        array.append(i)
    }
    
    for i in 2...num {
        if array[i] != 0 {
            var j = i + i
            while j <= num {
                array[j] = 0
                j += i
            }
        }
    }
    
    for i in 2...num {
        if array[i] != 0 {
            result.append(i)
        }
    }
    
    return result
}
print(getPrimeNumberEratostheneen(100))

4. 5. GCD Euclideon Algorithm

func GCD(_ n:Int,_ m:Int) -> Int {
  if n > m {
    if (n % m == 0) {
      return m
    } else {
      return GCD(m, n % m)
    }
  } else {
    if (m % n == 0) {
      return n
    } else {
      return GCD(n, m % n)
    }
  }
}
GCD(25,3)