# Euler problems/181 to 190

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Investigating in how many ways objects of two different colours can be grouped. | Investigating in how many ways objects of two different colours can be grouped. | ||

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== [http://projecteuler.net/index.php?section=problems&id=182 Problem 182] == | == [http://projecteuler.net/index.php?section=problems&id=182 Problem 182] == | ||

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Solution: | Solution: | ||

<haskell> | <haskell> | ||

− | fun a1 b1 = | + | fun a1 b1 = sum [ e | e <- [2..a*b-1], |

− | + | gcd e (a*b) == 1, | |

− | + | gcd (e-1) a == 2, | |

− | + | gcd (e-1) b == 2 ] | |

− | + | where a = a1-1 | |

− | + | b = b1-1 | |

− | + | ||

− | + | problem_182 = fun 1009 3643 | |

− | + | ||

− | + | ||

− | problem_182=fun 1009 3643 | + | |

</haskell> | </haskell> | ||

## Revision as of 22:24, 24 February 2008

## 1 Problem 181

Investigating in how many ways objects of two different colours can be grouped.

## 2 Problem 182

RSA encryption.

Solution:

fun a1 b1 = sum [ e | e <- [2..a*b-1], gcd e (a*b) == 1, gcd (e-1) a == 2, gcd (e-1) b == 2 ] where a = a1-1 b = b1-1 problem_182 = fun 1009 3643

## 3 Problem 183

Maximum product of parts.

Solution:

-- Does the decimal expansion of p/q terminate? terminating p q = 1 == reduce [2,5] (q `div` gcd p q) where reduce [] n = n reduce (x:xs) n | n `mod` x == 0 = reduce (x:xs) (n `div` x) | otherwise = reduce xs n -- The expression (round $ fromIntegral n / e) computes the integer k -- for which (n/k)^k is at a maximum. Also note that, given a rational number -- r and a natural number k, the decimal expansion of r^k terminates if -- and only if the decimal expansion of r does. answer = sum [if terminating n (round $ fromIntegral n / e) then -n else n | n <- [5 .. 10^4]] where e = exp 1 main = print answer