# Euler problems/1 to 10

### From HaskellWiki

m (Completed definition of primes in problem 7) |
m (Corrected links to the Euler project) |
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[[Category:Programming exercise spoilers]] | [[Category:Programming exercise spoilers]] | ||

− | == [http://projecteuler.net/index.php?section= | + | == [http://projecteuler.net/index.php?section=view&id=1 Problem 1] == |

Add all the natural numbers below 1000 that are multiples of 3 or 5. | Add all the natural numbers below 1000 that are multiples of 3 or 5. | ||

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

− | == [http://projecteuler.net/index.php?section= | + | == [http://projecteuler.net/index.php?section=view&id=2 Problem 2] == |

Find the sum of all the even-valued terms in the Fibonacci sequence which do not exceed one million. | Find the sum of all the even-valued terms in the Fibonacci sequence which do not exceed one million. | ||

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

− | == [http://projecteuler.net/index.php?section= | + | == [http://projecteuler.net/index.php?section=view&id=3 Problem 3] == |

Find the largest prime factor of 317584931803. | Find the largest prime factor of 317584931803. | ||

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

− | == [http://projecteuler.net/index.php?section= | + | == [http://projecteuler.net/index.php?section=view&id=4 Problem 4] == |

Find the largest palindrome made from the product of two 3-digit numbers. | Find the largest palindrome made from the product of two 3-digit numbers. | ||

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

− | == [http://projecteuler.net/index.php?section= | + | == [http://projecteuler.net/index.php?section=view&id=5 Problem 5] == |

What is the smallest number divisible by each of the numbers 1 to 20? | What is the smallest number divisible by each of the numbers 1 to 20? | ||

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

− | == [http://projecteuler.net/index.php?section= | + | == [http://projecteuler.net/index.php?section=view&id=6 Problem 6] == |

What is the difference between the sum of the squares and the square of the sums? | What is the difference between the sum of the squares and the square of the sums? | ||

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

− | == [http://projecteuler.net/index.php?section= | + | == [http://projecteuler.net/index.php?section=view&id=7 Problem 7] == |

Find the 10001st prime. | Find the 10001st prime. | ||

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

− | == [http://projecteuler.net/index.php?section= | + | == [http://projecteuler.net/index.php?section=view&id=8 Problem 8] == |

Discover the largest product of five consecutive digits in the 1000-digit number. | Discover the largest product of five consecutive digits in the 1000-digit number. | ||

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

− | == [http://projecteuler.net/index.php?section= | + | == [http://projecteuler.net/index.php?section=view&id=9 Problem 9] == |

There is only one Pythagorean triplet, {''a'', ''b'', ''c''}, for which ''a'' + ''b'' + ''c'' = 1000. Find the product ''abc''. | There is only one Pythagorean triplet, {''a'', ''b'', ''c''}, for which ''a'' + ''b'' + ''c'' = 1000. Find the product ''abc''. | ||

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

− | == [http://projecteuler.net/index.php?section= | + | == [http://projecteuler.net/index.php?section=view&id=10 Problem 10] == |

Calculate the sum of all the primes below one million. | Calculate the sum of all the primes below one million. | ||

## Revision as of 10:26, 20 July 2007

## Contents |

## 1 Problem 1

Add all the natural numbers below 1000 that are multiples of 3 or 5.

Solution:

problem_1 = sum [ x | x <- [1..999], (x `mod` 3 == 0) || (x `mod` 5 == 0)]

problem_1_v2 = sum $ filter (\x -> ( x `mod` 3 == 0 || x `mod` 5 == 0 ) ) [1..999]

## 2 Problem 2

Find the sum of all the even-valued terms in the Fibonacci sequence which do not exceed one million.

Solution:

problem_2 = sum [ x | x <- takeWhile (<= 1000000) fibs, x `mod` 2 == 0] where fibs = 1 : 1 : zipWith (+) fibs (tail fibs)

## 3 Problem 3

Find the largest prime factor of 317584931803.

Solution:

primes = 2 : filter ((==1) . length . primeFactors) [3,5..] primeFactors n = factor n primes where factor n (p:ps) | p*p > n = [n] | n `mod` p == 0 = p : factor (n `div` p) (p:ps) | otherwise = factor n ps problem_3 = last (primeFactors 317584931803)

## 4 Problem 4

Find the largest palindrome made from the product of two 3-digit numbers.

Solution:

problem_4 = foldr max 0 [ x | y <- [100..999], z <- [100..999], let x = y * z, let s = show x, s == reverse s]

## 5 Problem 5

What is the smallest number divisible by each of the numbers 1 to 20?

Solution:

problem_5 = head [ x | x <- [2520,5040..], all (\y -> x `mod` y == 0) [1..20]]

An alternative solution that takes advantage of the Prelude to avoid use of the generate and test idiom:

problem_5' = foldr1 lcm [1..20]

## 6 Problem 6

What is the difference between the sum of the squares and the square of the sums?

Solution:

problem_6 = sum [ x^2 | x <- [1..100]] - (sum [1..100])^2

## 7 Problem 7

Find the 10001st prime.

Solution:

primes = 2 : filter ((==1) . length . primeFactors) [3,5..] primeFactors n = factor n primes where factor n (p:ps) | p*p > n = [n] | n `mod` p == 0 = p : factor (n `div` p) (p:ps) | otherwise = factor n ps problem_7 = head $ drop 10000 primes

## 8 Problem 8

Discover the largest product of five consecutive digits in the 1000-digit number.

Solution:

num = ... -- 1000 digit number as a string digits = map digitToInt num groupsOf _ [] = [] groupsOf n xs = take n xs : groupsOf n ( tail xs ) problem_8 = maximum . map product . groupsOf 5 $ digits

## 9 Problem 9

There is only one Pythagorean triplet, {*a*, *b*, *c*}, for which *a* + *b* + *c* = 1000. Find the product *abc*.

Solution:

problem_9 = head [a*b*c | a <- [1..500], b <- [a..500], let c = 1000-a-b, a^2 + b^2 == c^2]

## 10 Problem 10

Calculate the sum of all the primes below one million.

Solution:

problem_10 = sum (takeWhile (< 1000000) primes)