Project Euler – Problem 5 Solution

Problem

2520 is the smallest number that can be divided by each of the numbers from 1 to 10 without any remainder.

What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20?

Solution

let isEvenlyDivided(n, m) = n % m = 0
let isEvenlyDividedByAll(n, numbers) = numbers |> Seq.forall (fun x -> isEvenlyDivided(n, x))

let findSmallestCommonMultiple(numbers) =
    let max = Array.max(numbers)
    Seq.unfold (fun x -> Some(x, x + 1)) max
    |> Seq.filter (fun x -> isEvenlyDividedByAll(x, numbers))
    |> Seq.head

let commonMultiplier = findSmallestCommonMultiple [|1..20|]

Again, I build two functions to handle the logic of checking whether a number can be evenly divided by all the numbers in a supplied list. I have used another new function Seq.forall (I can’t find the MSDN doc for this function, but see List.forall instead) which tests each element in the list with the supplied predicate.

In order to find the smallest common multiple for all the numbers from 1 to 20, I first generated a sequence of all the natural numbers equal or greater than 20:

Seq.unfold (fun x –> Some(x, x + 1)) max // max is resolved to 20 by Array.max(numbers)

Then for each number I applied the predicate isEvenlyDividedByAll to see if it can be evenly divided by each of the numbers from 1 to 20, and finally, Seq.head returns the first element that matches the predicate and that’s our answer!

One last thing though, when you run this code you’ll see that it takes a rather long time to return. So to improve on the performance of this logic, you can add a small and yet effective step to only test numbers which are multiples of the largest number (20 in our case) in the list:

let findSmallestCommonMultiple(numbers) =
    let max = Array.max(numbers)
    Seq.unfold (fun x -> Some(x, x + 1)) 1
    |> Seq.map (fun x -> x * max)
    |> Seq.filter (fun x -> isEvenlyDividedByAll(x, numbers))
    |> Seq.head

Run the new findSmallestCommonMultiple function again and it now returns much much quicker :-)

  • bill

    hello, how long did the first solution take? I am still waiting for the output :). The second solution worked pretty fast.

  • theburningmonk

    @bill – that’s the beauty of simple optimization vs brute force approach ;-)

  • Omu

    actually you don’t have to check for all numbers, if a number can be divided to 20 => it can be div to 2,5 and 10 as well
    here’s my solution:

    let p5 =
    let isDivisor n =
    let rec loop = function
    | 10 -> true
    | x when n % x = 0 -> loop (x-1)
    | _ -> false
    loop 20
    let rec search = function
    | x when isDivisor x -> x
    | x -> search (x+1)
    search 2520

  • David Brower

    Excellent series of blog posts!

    I was able to get down to 8 seconds by generating just multiples of 20:

    let isDivisibleBy seq n =
    seq
    |> Seq.forall (fun x -> n % x = 0)

    Seq.unfold (fun x -> Some(x, x + 20)) 20
    |> Seq.find (isDivisibleBy [|1 .. 20|])