Project Euler, problem 26

Problem is stated as following:

---

A unit fraction contains 1 in the numerator. The decimal representation of the unit 
fractions with denominators 2 to 10 are given:

    1/2  =  0.5
    1/3  =  0.(3)
    1/4  =  0.25
    1/5  =  0.2
    1/6  =  0.1(6)
    1/7  =  0.(142857)
    1/8  =  0.125
    1/9  =  0.(1)
    1/10 =  0.1

Where 0.1(6) means 0.166666..., and has a 1-digit recurring cycle. It can be seen
that 1/7 has a 6-digit recurring cycle.

Find the value of d < 1000 for which 1/d contains the longest recurring cycle in 
its decimal fraction part.

---

Solution is based on idea that longest recurring cycle should have 1/p form, where p - prime number. After quick look at wikipedia about repeating decimals I thought that I should just search biggest prime, but at closer examination of wiki, I understood that not necessary biggest prime would result in biggest recurring cycle of 1/prime. So for the sake of mathematical integrity, I decided to implement 'correct' search of biggest recurring cycle, which is based on calculating multiplicative order of 10 modulo prime. Solution also is pretty fast - executes in about 130 ms. Below is solution code in language F#:

---

open System.Diagnostics

// http://en.wikipedia.org/wiki/Modular_exponentiation
let modular_exponent b e m =
    let rec modular_exponent0 b e e0 m c0 = 
        match () with
        | _ when e0 >= e -> c0
        | _  -> let e1 = e0 + 1 in
                let c1 = (c0 * b) % m in
                modular_exponent0 b e e1 m c1
    modular_exponent0 b e 0 m 1

// http://en.wikipedia.org/wiki/Multiplicative_order
let multiplicative_order a n =
    let rec multiplicative_order0 a n k0 = 
        let me = modular_exponent a k0 n in
            match me with
            | 1 -> k0
            | _ -> let k1 = k0 + 1 in
                   multiplicative_order0 a n k1
    multiplicative_order0 a n 1

// http://en.wikipedia.org/wiki/Repeating_decimal#Fractions_with_prime_denominators
let period_of_1divp p = multiplicative_order 10 p

// http://en.wikipedia.org/wiki/Prime_number#Verifying_primality
let is_prime x = 
    let rec is_prime0 x d0 = 
        match () with
        | _ when d0 > int (sqrt (float x)) -> true
        | _ when x % d0 = 0 -> false
        | _ -> let d1 = d0 + 1 in
               is_prime0 x d1
    is_prime0 x 2

let main = let stopwatch = new Stopwatch()
           stopwatch.Start()
           let prime,max_period = 
               [ for x in 6..999 do if is_prime x then yield x,period_of_1divp x ] |> 
               Seq.fold (fun (a,b) (x,y) -> if y > b then (x,y) else (a,b)) (0,0)
           stopwatch.Stop()
           printfn "Fraction with greatest period of %d is 1/%d, calculated in %d ms" 
                    max_period
                    prime
                    stopwatch.ElapsedMilliseconds

---

Have fun with Project Euler and F# !