Problem:
Score : 300 points

Problem Statement

You are given an integer N.

For two positive integers A and B, we will define F(A,B) as the larger of the following: the number of digits in the decimal notation of A, and the number of digits in the decimal notation of B.

For example, F(3,11) = 2 since 3 has one digit and 11 has two digits.

Find the minimum value of F(A,B) as (A,B) ranges over all pairs of positive integers such that N = A \times B.

Constraints

1 \leq N \leq 10^{10}

N is an integer.

Input

The input is given from Standard Input in the following format:

N

Output

Print the minimum value of F(A,B) as (A,B) ranges over all pairs of positive integers such that N = A \times B.

Sample Input 1

10000

Sample Output 1

3

F(A,B) has a minimum value of 3 at (A,B)=(100,100).

Sample Input 2

1000003

Sample Output 2

7

There are two pairs (A,B) that satisfy the condition: (1,1000003) and (1000003,1). For these pairs, F(1,1000003)=F(1000003,1)=7.

Sample Input 3

9876543210

Sample Output 3

6

Solution in Haskell (GHC 7.10.3) (judge verdict: Runtime Error)

import qualified Data.Map as M
import Data.List (sort)
type Primes = M.Map Integer Integer
type Dim = [Integer]
type Coord = [Integer]
type Base = [Integer]
type Index = Integer

main :: IO ()
main = do
    n <- readLn
    print $ solve n

solve :: Integer -> Int
solve n = minimum $ map f as
    where
        f a = max (length $ show a) (length $ show $ n `div` a)
        as = takeWhile g . sort $ map (eval dim base) [1 .. m]
        g x = (floor' . sqrt . fromIntegral $ n) > x
        ps0 = primes n
        dim = map (+1) $ M.elems ps0
        base = M.keys ps0
        m = product dim

floor' :: Double -> Integer
floor' = floor

eval :: Dim -> Base -> Index -> Integer
eval dim base i = product $ zipWith (^) base coord
    where coord = index2coord dim i

index2coord :: Dim -> Index -> Coord
index2coord dim i = tail . map snd $ scanl f (i,0) dim
    where f (a,_) x = (a `div` x, a `mod` x)

primes :: Integer -> Primes
primes 1 = M.empty
primes n = M.unionWith (+) p1 (primes n')
    where
        v = factor1 n
        p1 = M.singleton v 1
        n' = n `div` v

factor1 :: Integer -> Integer
factor1 n = head ks
    where ks = [x | x <- 2:[3,5..], n `mod` x == 0]