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 Ruby (2.7.1) (judge verdict: Wrong Answer)

require 'prime'
n = gets.to_i
nums = gets.split.map(&:to_i).sort
hash = Hash.new(0)
gcd = nums.first
set = false
time = Time.now
nums.shuffle!
interrupted = false
nums.each_with_index do |num, index|
  gcd = gcd.gcd(num)
  next if interrupted
  if index % 1000 == 0
    unless Time.now - 1.8 < time
      interrupted = true
    end
  end
  unless set
    Prime.prime_division(num).each do |k, _|
      hash[k] += 1
      set = true if hash[k] > 1
    end
  end
end
if gcd > 1
  puts "not coprime"
else
  if set
    puts "setwise coprime"
  else
    puts "pairwise coprime"
  end
end