Comments on: Project Euler in F# – Problem 5

By: Project Euler in F# - Problem 5 (alternative solution) » FSharp.it

Project Euler in F# - Problem 5 (alternative solution) » FSharp.it — Fri, 08 Feb 2008 14:57:01 +0000

[...] was some buzz concerning my last post on Project Euler Problem 5, since the solution presented implemented a naive brute-force algorithm [...]

By: Paolo Bonzini

Paolo Bonzini — Fri, 08 Feb 2008 09:54:23 +0000

gcd(a,b)*lcm(a,b) = a*b holds always. gcd(a,b,c)*lcm(a,b,c) = a*b*c does not hold, but you can exploit the associativity of lcm like this:

lcm(a,b,c)=lcm(lcm(a,b),c)

and then use the gcd(a,b)*lcm(a,b) = a*b identity to compute lcm.

in Smalltalk:

st> Time millisecondsToRun: [ (2 to: 20) fold: [ :a :b | a lcm: b ] ]
1
st> Time millisecondsToRun: [ (2 to: 1000) fold: [ :a :b | a lcm: b ] ]
14

and that’s a 433-digit number…

By: Michaelk

Michaelk — Fri, 08 Feb 2008 01:16:02 +0000

I noticed that the LCM of 1..18 was a factor of the LCM of 1..19, and the LCM of 1..19 was a factor of 1..20.

So, I just combined that with memoization, and got it working in Python (real 0m0.012s).

By: Michael

Michael — Thu, 07 Feb 2008 17:01:23 +0000

claudio
if you think about it the binary gcd(a,b)*lm(a,b) = a*b doesn’t hold in the general case. gcd(1,..,20) should obviously be 1 and lcm(1,..,20) is obviously not 1*2*…*20 which means lcm(1,..,20) \not= 1*…*20/gcd(1,..,20)

But lcm is still easy it’s just the max of each exponent of the unique factorization of each of the number you take lcm over.

By: Matt

Matt — Thu, 07 Feb 2008 17:00:43 +0000

Here’s my solution with no programming required
http://methodoverload.blogspot.com/2008/02/project-euler-5-and-number-theory.html

By: claudio

claudio — Thu, 07 Feb 2008 16:53:44 +0000

Hi, thanks for the suggestion.

I plan to write a new post soon presenting your solution and showing the huge difference between a brute-force approach and a “smart” one.

The goal of this series of articles is to introduce people to functional programming, trying to translate the basic algorithms that come to mind into F# code, and hoping that the readers will try to find their own solution before I show mine.

BTW, I also added the comment subscription plugin that you suggested, thanks!

By: Beetle B.

Beetle B. — Thu, 07 Feb 2008 16:35:22 +0000

If you know a bit more math, this is even easier:

lcm(a1, a2, …) = (a1*a2*a3*…)/gcd(a1,a2,…)

where gcd is the greatest common divisor and can be calculated using Euclid’s algorithm.

Basically, I calculated the lcm of 2 and 3, and then the lcm of the result and 4, the lcm of the result and 5, etc.

0.1 ms in Python (interpreted!) on a P4 2.53 GHz machine.

Oh, and BTW, since you’re using Wordpress, you may want to get the plugin that allows people who post comments to subscribe to the comments via email (yes, I know you have an RSS, but I don’t want to crowd my RSS aggregator for simple things like these). Otherwise, it’ll be hard for commenters to maintain a conversation with you.