Problem:
Score : 500 points

Problem Statement

We have N integers. The i-th number is A_i.

\{A_i\} is said to be pairwise coprime when GCD(A_i,A_j)=1 holds for every pair (i, j) such that 1\leq i < j \leq N.

\{A_i\} is said to be setwise coprime when \{A_i\} is not pairwise coprime but GCD(A_1,\ldots,A_N)=1.

Determine if \{A_i\} is pairwise coprime, setwise coprime, or neither.

Here, GCD(\ldots) denotes greatest common divisor.

Constraints

2 \leq N \leq 10^6

1 \leq A_i\leq 10^6

Input

Input is given from Standard Input in the following format:

N
A_1 \ldots A_N

Output

If \{A_i\} is pairwise coprime, print pairwise coprime; if \{A_i\} is setwise coprime, print setwise coprime; if neither, print not coprime.

Sample Input 1

3
3 4 5

Sample Output 1

pairwise coprime

GCD(3,4)=GCD(3,5)=GCD(4,5)=1, so they are pairwise coprime.

Sample Input 2

3
6 10 15

Sample Output 2

setwise coprime

Since GCD(6,10)=2, they are not pairwise coprime. However, since GCD(6,10,15)=1, they are setwise coprime.

Sample Input 3

3
6 10 16

Sample Output 3

not coprime

GCD(6,10,16)=2, so they are neither pairwise coprime nor setwise coprime.

Solution in Haskell (GHC 8.8.3) (judge verdict: Accepted)

import           Control.Monad
import           Control.Monad.ST
import           Data.Array
import           Data.Array.ST
import qualified Data.ByteString.Char8 as BS
import           Data.Maybe
import qualified Data.Vector           as V
import qualified Data.Vector.Mutable   as M

readInt = fst . fromJust . BS.readInt
readIntList = map readInt . BS.words
getIntList = readIntList <$> BS.getLine

main=do
    getLine
    a <- getIntList
    let cnt = runST $ do
            vec <- V.thaw $ V.replicate m 0
            forM_ a $ \ai ->
                forM_ (primeFactor ai) $ \j ->
                    M.modify vec (+1) j
            V.freeze vec
    putStrLn$
        if V.and $V.map (<=1) cnt
        then "pairwise coprime"
        else if foldl1 gcd a == 1
            then  "setwise coprime"
            else  "not coprime"
m = 10^6+1

table :: V.Vector Int
table = runST $ do
    vec <- V.thaw $ V.fromList [0..m]
    forM_ (takeWhile (\i -> i*i < m) [2..]) $ \i -> do
        arri <- M.read vec i
        when (arri >= i)$
            forM_ [i,i+i..m-1] $ \j -> do
                v <- M.read vec j
                when (v == j) $ M.write vec j i
    V.freeze vec

primeFactor :: Int -> [Int]
primeFactor 1 = []
primeFactor n = if null prev || head prev /= p
    then p:prev
    else prev
    where p = table V.! n
          prev = primeFactor (n `div` p)