struct Rational {
	var numerator: Int?
	var denominator: Int?

	var common: Int? {
		if let n = numerator, let d = denominator {
			return gcd(a: n, b: d)
		}
		else {
			return nil
		}
	}

	var simple: Rational {
		get {
			if let n = numerator, let d = denominator, let c = common {
				return Rational(n: n / c, d: d / c)
			}
			else {
				return Rational(n: nil, d: nil)
			}
		}
	}

	var reciprocal: Rational {
		get {
			return Rational(n: denominator, d: numerator)
		}
		set(reci) {
			numerator = reci.denominator
			denominator = reci.numerator
		}
	}

	init(n: Int?, d: Int?) {
		if let validNumerator = n, let validDenominator = d {
			if validDenominator != 0 {
				self.numerator = validNumerator
				self.denominator = validDenominator
			}
			else {
				self.numerator = nil
				self.denominator = nil
			}
		}
		else {
			self.numerator = nil
			self.denominator = nil
		}
	}

	func isBigger(than comparison: Rational) -> Bool {
		if let n = numerator, let cn = comparison.denominator, let d = denominator, let cd = denominator {
			return n * cd > d * cn
		}
		else {
			return false
		}
	}

	func isEqual(to comparison: Rational) -> Bool {
		if let n = numerator, let cn = comparison.denominator, let d = denominator, let cd = denominator {
			return n * cd == d * cn
		}
		else {
			return false
		}
	}

	func isSmaller(than comparison: Rational) -> Bool {
		if let n = numerator, let cn = comparison.denominator, let d = denominator, let cd = denominator {
			return n * cd < d * cn
		}
		else {
			return false
		}
	}

	func added(to addition: Rational) -> Rational {
		if let n = numerator, let an = addition.numerator, let d = denominator, let ad = addition.denominator {
			return Rational(n: n * ad + d * an ,d: d * ad) // 2/3 + 5/6 = 12/18 + 15/18
	}
		else{
			return Rational(n: nil, d: nil)
		}
	}

	func subtracted(from subtraction: Rational) -> Rational {
		if let n = numerator, let sn = subtraction.numerator, let d = denominator, let sd = subtraction.denominator {
			return Rational(n: n * sd - d * sn,d: d * sd)
		}
		else {
			return Rational(n: nil, d: nil)
		}
	}

	func multiplied(by multiplier: Rational) -> Rational {
		if let n = numerator, let mn = multiplier.numerator, let d = denominator, let md = multiplier.denominator {
			return Rational(n: n * mn ,d: d * md)
		}
		else {
			return Rational(n: nil, d: nil)
		}
	}

	func devided(by devider: Rational) -> Rational {
		if let n = numerator, let dn = devider.numerator, let d = denominator, let dd = devider.denominator {
			return Rational(n: n * dd ,d: d * dn)
		}
		else {
			return Rational(n: nil, d: nil)
		}
	}
}

func gcd(a: Int, b: Int) -> Int {
	var i = a
	var j = b
	while true {
		i %= j
		(i, j) = (j, i)
		if j == 0 {
			return i
		}
	}
}

func lcm(a: Int, b: Int) -> Int {
	return a * b / gcd(a: a, b: b)
}