13 May 2008

Problem #5:

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

Speaking of infinite sequences, here’s the solution that seemed obvious to me at first.

```
open Microsoft.FSharp.Math
Seq.init_infinite (fun x > BigInt.FromInt32 x)
|> Seq.find(fun (x:BigInt) ->
x > 1I && Seq.for_all (fun (n:BigInt) -> x % n = 0I) [1I..20I])
```

Nice and simple, just iterate through all numbers, and find the first ne that’s evenly divisible by all the numbers from 1 to 20. Unfortunately let it run for an hour (!) while I had dinner, and it hadn’t completed. I just assumed was doing something dumb, but it seems to give sensible answers for the numbers in the range of 1..10 and 1..16. Apparently, at 1..17 or more though the answer becomes “quite” large. The first version just used int32’s rather than BigInt; I thought perhaps it was getting past 2^32 and so never finding the answer, but it’s hard to say since I wasn’t prepared to wait any longer for the BigInt version to finish.

As a side note, the bigint stuff is a bit ugly in this example, I guess an unfortunate side effect of .NET showing through where the numerical stack is fractured (sensibly and everything, just a little unfortunate here).

In any case, a more mathematical and less brute-force algorithmic approach seems to be required for this problem.

Here’s one that takes only milliseconds to run based on:

http://en.wikipedia.org/wiki/Least_common_multiple#Alternative_method

```
let rec numTimes(x, p) =
if x % p = 0 then numTimes(x / p, p) + 1
else 0
let maxNumTimes p =
let num = float([1..20] |> Seq.map(fun x -> numTimes(x, p)) |> Seq.fold(max) 0)
int(Math.Pow(float(p), num))
[1..20]
|> Seq.filter(isprime)
|> Seq.map(maxNumTimes)
|> Seq.fold1( * )
```

For all primes in the range, figure out the maximum number of times that prime appears in the prime factorization of any of the numbers, and find the product of those primes raised to the maximum number of powers. I learned a little F# in this one, but as far as the math, it would have taken me a long while to recall or figure that out, so it was basically just implementing what Wikipedia described. Oh well.

My “fold” got one better again, instead of passing a pointless anon function and a pointless initial value, last time I got rid of the function. This time I found “fold1” and got rid of the non-useful initial value too! :)