Nastia and a Good Array (Codeforces 1521B)
Tags: Math, number theory

Nastia has received an array of $$$n$$$ positive integers as a gift.She calls such an array $$$a$$$ good that for all $$$i$$$ ($$$2 \le i \le n$$$) takes place $$$gcd(a_{i - 1}, a_{i}) = 1$$$, where $$$gcd(u, v)$$$ denotes the greatest common divisor (GCD) of integers $$$u$$$ and $$$v$$$.You can perform the operation: select two different indices $$$i, j$$$ ($$$1 \le i, j \le n$$$, $$$i \neq j$$$) and two integers $$$x, y$$$ ($$$1 \le x, y \le 2 \cdot 10^9$$$) so that $$$\min{(a_i, a_j)} = \min{(x, y)}$$$. Then change $$$a_i$$$ to $$$x$$$ and $$$a_j$$$ to $$$y$$$.The girl asks you to make the array good using at most $$$n$$$ operations.It can be proven that this is always possible.

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 length of the array.The second line of each test case contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_{n}$$$ ($$$1 \le a_i \le 10^9$$$) — the array which Nastia has received as a gift.It's guaranteed that the sum of $$$n$$$ in one test doesn't exceed $$$2 \cdot 10^5$$$.

Output: For each of $$$t$$$ test cases print a single integer $$$k$$$ ($$$0 \le k \le n$$$) — the number of operations. You don't need to minimize this number.In each of the next $$$k$$$ lines print $$$4$$$ integers $$$i$$$, $$$j$$$, $$$x$$$, $$$y$$$ ($$$1 \le i \neq j \le n$$$, $$$1 \le x, y \le 2 \cdot 10^9$$$) so that $$$\min{(a_i, a_j)} = \min{(x, y)}$$$ — in this manner you replace $$$a_i$$$ with $$$x$$$ and $$$a_j$$$ with $$$y$$$.If there are multiple answers, print any.

Note: Consider the first test case.Initially $$$a = [9, 6, 3, 11, 15]$$$.In the first operation replace $$$a_1$$$ with $$$11$$$ and $$$a_5$$$ with $$$9$$$. It's valid, because $$$\min{(a_1, a_5)} = \min{(11, 9)} = 9$$$.After this $$$a = [11, 6, 3, 11, 9]$$$.In the second operation replace $$$a_2$$$ with $$$7$$$ and $$$a_5$$$ with $$$6$$$. It's valid, because $$$\min{(a_2, a_5)} = \min{(7, 6)} = 6$$$.After this $$$a = [11, 7, 3, 11, 6]$$$ — a good array.In the second test case, the initial array is already good.

Sample Inputs:
2
5
9 6 3 11 15
3
7 5 13

Sample Outputs:
2
1 5 11 9
2 5 7 6
0

Solution (Python):
def isprime(n):
    if n <= 1:
        return False
    if n <= 3:
        return True
    if n % 2 == 0 or n % 3 == 0:
        return False
    i = 5
    while i * i <= n:
        if n % i == 0 or n % (i + 2) == 0:
            return False
        i += 6
    return True
maxi = 2000000000
while True:
    if isprime(maxi):
        break
    maxi -= 1
for i in range(int(input())):
    n = int(input())
    l = list(map(int, input().split()))
    c = 0
    s = ''
    j = 0
    while j < n:
        if l[j] != maxi and j < n - 1:
            s += f'{j + 1} {j + 2} {min(l[j], l[j + 1])} {maxi}\n'
            j += 2
            c += 1
        elif l[j] == maxi and j > 0:
            s += f'{j} {j + 1} {min(l[j - 1], l[j])} {maxi + 1}\n'
            j += 1
            c += 1
        else:
            j += 1
    print(c)
    print(s.strip())

Solution (Python):
def solve(n, a):
    swaps = 0
    ans = []
    for j in range(1, n, 2):
        i = j - 1
        x = min(a[i], a[j])
        y = 10 ** 9 + 7
        ans.append([i + 1, j + 1, x, y])
        swaps += 1
    print(swaps)
    for move in ans:
        print(' '.join(map(str, move)))
if __name__ == '__main__':
    t = int(input())
    for _ in range(t):
        n = int(input().strip())
        a = list(map(int, input().strip().split()))
        solve(n, a)

Solution (Python):
out = []
for _ in range(int(input())):
    n = int(input())
    data = [x for x in map(int, input().split())]
    minv = min(data)
    pos = data.index(minv)
    out.append('{}'.format(n - 1))
    for i in range(len(data)):
        if i != pos:
            out.append('{} {} {} {}'.format(pos + 1, i + 1, minv, minv + abs(i - pos)))
print('\n'.join(out))