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: Wrong Answer)

{-# LANGUAGE BangPatterns #-}
import           Control.Exception           (assert)
import           Control.Monad
import           Control.Monad.Primitive
import qualified Control.Monad.ST            as ST
import qualified Data.Array.IO               as IO
import qualified Data.Array.ST               as ST
import qualified Data.Array.Unboxed          as A
import           Data.Bits
import qualified Data.ByteString.Char8       as BS
import           Data.Char
import           Data.Foldable
import           Data.List
import qualified Data.Map.Strict             as Map
import           Data.Maybe
import qualified Data.Sequence               as Seq
import qualified Data.Set                    as Set
import qualified Data.Vector                 as VB
import qualified Data.Vector.Mutable         as VBM
import qualified Data.Vector.Unboxed         as V
import qualified Data.Vector.Unboxed.Mutable as VM
import           Debug.Trace

readInt = fst . fromJust . BS.readInt
readIntList = map readInt . BS.words
getInt = readInt <$> BS.getLine
getIntList = readIntList <$> BS.getLine
getIntVec n = V.unfoldrN n (BS.readInt . BS.dropWhile isSpace) <$> BS.getLine

readInteger = fst . fromJust . BS.readInteger
readIntegerList = map readInteger . BS.words
getInteger = readInteger <$> BS.getLine
getIntegerList = readIntegerList <$> BS.getLine

inf :: Int
inf = 10^18

main = do
  n <- getInt
  as <- getIntVec n

  d1 <- VM.new (10^6+1)
  VM.set d1 (0::Int)
  d2 <- VM.new (10^6+1)
  VM.set d2 (0::Int)

  let sieve :: Int -> IO ()
      sieve i | i*i > 10^6 = return ()
              | otherwise = do
                  x <- VM.read d1 i
                  when (x == 0) $
                    forM_ [2..(10^6 `div` i)] $ \j -> do
                    let k = i * j
                    y <- VM.read d1 k
                    when (y == 0) $ VM.write d1 k i
                  sieve (i+1)

  sieve 2

  let pfactr :: Int -> Set.Set Int -> IO [Int]
      pfactr x s = do y <- VM.read d1 x
                      if y == 0
                        then return $ Set.toList (Set.insert x s)
                        else pfactr (x `div` y) (Set.insert y s)

      loop :: [Int] -> Int -> IO Int
      loop [] res = return res
      loop (f:fs) res = do
        t <- VM.read d2 f
        VM.write d2 f (t+1)
        let res' = max res (t+1)
        loop fs res'

      solve :: Int -> Int -> IO Int
      solve i res | i >= n = return res
                  | otherwise = do
                      let a = as V.! i
                      fs <- pfactr a Set.empty
                      res' <- loop fs res
                      solve (i+1) res'

  x <- solve 0 0
  let ans | x == 1 = "pairwise coprime"
          | x < n  = "setwise coprime"
          | otherwise = "not coprime"

  putStrLn ans