Di-visible Confusion (Codeforces 1603A)
Tags: Math, number theory

YouKn0wWho has an integer sequence $$$a_1, a_2, \ldots, a_n$$$. He will perform the following operation until the sequence becomes empty: select an index $$$i$$$ such that $$$1 \le i \le |a|$$$ and $$$a_i$$$ is not divisible by $$$(i + 1)$$$, and erase this element from the sequence. Here $$$|a|$$$ is the length of sequence $$$a$$$ at the moment of operation. Note that the sequence $$$a$$$ changes and the next operation is performed on this changed sequence.For example, if $$$a=[3,5,4,5]$$$, then he can select $$$i = 2$$$, because $$$a_2 = 5$$$ is not divisible by $$$i+1 = 3$$$. After this operation the sequence is $$$[3,4,5]$$$.Help YouKn0wWho determine if it is possible to erase the whole sequence using the aforementioned operation.

Input: The first line contains a single integer $$$t$$$ ($$$1 \le t \le 10\,000$$$)  — the number of test cases.The first line of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 10^5$$$).The second line of each test case contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \le a_i \le 10^9$$$).It is guaranteed that the sum of $$$n$$$ over all test cases doesn't exceed $$$3 \cdot 10^5$$$.

Output: For each test case, print "YES" (without quotes) if it is possible to erase the whole sequence using the aforementioned operation, print "NO" (without quotes) otherwise. You can print each letter in any register (upper or lower).

Note: In the first test case, YouKn0wWho can perform the following operations (the erased elements are underlined): $$$[1, \underline{2}, 3] \rightarrow [\underline{1}, 3] \rightarrow [\underline{3}] \rightarrow [\,].$$$In the second test case, it is impossible to erase the sequence as $$$i$$$ can only be $$$1$$$, and when $$$i=1$$$, $$$a_1 = 2$$$ is divisible by $$$i + 1 = 2$$$.

Sample Inputs:
5
3
1 2 3
1
2
2
7 7
10
384836991 191890310 576823355 782177068 404011431 818008580 954291757 160449218 155374934 840594328
8
6 69 696 69696 696969 6969696 69696969 696969696

Sample Outputs:
YES
NO
YES
YES
NO

Solution (Python):
from math import gcd

def s():
    n = [int(i) for i in input().split()][0]
    a = [int(i) for i in input().split()]
    (l, r) = (2, 'YES\n')
    n = min(n, 28)
    for j in range(n):
        if a[j] % l == 0:
            r = 'NO\n'
            break
        l = l * (j + 3) // gcd(j + 3, l)
    return r
p = ''
for _ in range(int(input())):
    p += s()
print(p)

Solution (Python):
import sys
input = sys.stdin.readline
import math
''
t = int(input())
for _ in range(t):
    n = int(input())
    a = list(map(int, input().split()))
    flag = 1
    for i in range(min(22, n)):
        k = 2
        while k <= i + 2:
            if a[i] % k != 0:
                flag = 0
                break
            k += 1
        else:
            flag = 1
            break
    if flag == 0:
        print('YES')
    else:
        print('NO')

Solution (Python):
import sys
input = sys.stdin.readline
from math import gcd

def s():
    n = [int(i) for i in input().split()][0]
    a = [int(i) for i in input().split()]
    (l, r) = (2, 'YES\n')
    n = min(n, 28)
    for j in range(n):
        if a[j] % l == 0:
            r = 'NO\n'
            break
        l = l * (j + 3) // gcd(j + 3, l)
    return r
p = ''
for _ in range(int(input())):
    p += s()
print(p)