Euler problems/1 to 10
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1 Problem 1
Add all the natural numbers below 1000 that are multiples of 3 or 5.
Two solutions usingimport Data.List (union) problem_1' = sum (union [3,6..999] [5,10..999]) problem_1 = sum [x  x < [1..999], x `mod` 3 == 0  x `mod` 5 == 0]
Another solution which uses algebraic relationships:
problem_1 = sumStep 3 999 + sumStep 5 999  sumStep 15 999 where sumStep s n = s * sumOnetoN (n `div` s) sumOnetoN n = n * (n+1) `div` 2
2 Problem 2
Find the sum of all the evenvalued terms in the Fibonacci sequence which do not exceed one million.
Solution:
problem_2 = sum [ x  x < takeWhile (<= 1000000) fibs, even x] where fibs = 1 : 1 : zipWith (+) fibs (tail fibs)
The following two solutions use the fact that the evenvalued terms in the Fibonacci sequence themselves form a Fibonaccilike sequence that satisfies
problem_2 = sumEvenFibs $ numEvenFibsLessThan 1000000 where sumEvenFibs n = (evenFib n + evenFib (n+1)  2) `div` 4 evenFib n = round $ (2 + sqrt 5) ** (fromIntegral n) / sqrt 5 numEvenFibsLessThan n = floor $ (log (fromIntegral n  0.5) + 0.5*log 5) / log (2 + sqrt 5)
The first two solutions work because 10^6 is small. The following solution also works for much larger numbers (up to at least 10^1000000 on my computer):
problem_2 = sumEvenFibsLessThan 1000000 sumEvenFibsLessThan n = (a + b  1) `div` 2 where n2 = n `div` 2 (a, b) = foldr f (0,1) . takeWhile ((<= n2) . fst) . iterate times2E $ (1, 4) f x y  fst z <= n2 = z  otherwise = y where z = x `addE` y addE (a, b) (c, d) = (a*d + b*c  4*ac, ac + b*d) where ac=a*c times2E (a, b) = addE (a, b) (a, b)
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)
Another solution, not using recursion, is:
problem_3 = (m !! 0) `div` (m !! 1) where m = reverse $ takeWhile (<=n) (scanl1 (*) [ x  x < 2:[3,5..], (n `mod` x) == 0 ]) n = 600851475143
4 Problem 4
Find the largest palindrome made from the product of two 3digit numbers.
Solution:
problem_4 = maximum [x  y<[100..999], z<[y..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 = 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 [1..100])^2  sum (map (^2) [1..100])
7 Problem 7
Find the 10001st prime.
Solution:
primes in problem_3 problem_7 = primes !! 10000
8 Problem 8
Discover the largest product of five consecutive digits in the 1000digit number.
Solution:
import Data.Char (digitToInt) import Data.List (tails) problem_8 = do str < readFile "number.txt"  This line just converts our str(ing) to a list of 1000 Ints let number = map digitToInt (concat $ lines str) print $ maximum $ map (product . take 5) (tails number)
9 Problem 9
There is only one Pythagorean triplet, {a, b, c}, for which a + b + c = 1000. Find the product abc.
Solution:
triplets l = [[a,b,c]  m < [2..limit], n < [1..(m1)], let a = m^2  n^2, let b = 2*m*n, let c = m^2 + n^2, a+b+c==l] where limit = floor . sqrt . fromIntegral $ l problem_9 = product . head . triplets $ 1000
10 Problem 10
Calculate the sum of all the primes below one million.
Solution:
primes in problem_3 problem_10 = sum (takeWhile (< 1000000) primes)