# Difference between revisions of "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

## Problem 181

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

## 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
```

## 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
```