# Difference between revisions of "Euler problems/1 to 10"

(→[http://projecteuler.net/index.php?section=problems&id=5 Problem 5]: clarify remark) |
(→[http://projecteuler.net/index.php?section=problems&id=3 Problem 3]: bug fix (previously found largest proper divisor 67*3919, not prime)) |
||

Line 21: | Line 21: | ||

Solution: |
Solution: |
||

<haskell> |
<haskell> |
||

− | problem_3 = maximum [ x | x <- [1..round $ sqrt (fromInteger c)], c `mod` x == 0] |
||

+ | primes = 2 : filter ((==1) . length . primeFactors) [3,5..] |
||

− | where c = 317584931803 |
||

+ | 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 = maximum (primeFactors 317584931803) |
||

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

## Revision as of 17:51, 27 March 2007

## Contents

## 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 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)
```

## 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 = maximum (primeFactors 317584931803)
```

## 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]
```

## 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]
```

## 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
```

## Problem 7

Find the 10001st prime.

Solution:

```
problem_7 = head $ drop 10000 primes
where primes = 2:3:..
```

## Problem 8

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

Solution:

```
problem_8 = undefined
```

## 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]
```

## Problem 10

Calculate the sum of all the primes below one million.

Solution:

```
problem_10 = sum (takeWhile (< 1000000) primes)
```