Sequence GCD WGCD Solution Codechef

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Codechef Sequence GCD WGCD Solution

You are given an integer sequence A=(A1,A2,…,AN)A=(A1,A2,…,AN) of length NN and an integer MM such that 0≤M<∑i=1NAi0≤M<∑i=1NAi.

An integer sequence B=(B1,B2,…,BN)B=(B1,B2,…,BN) of length NN is called good if:

0≤Bi≤Ai0≤Bi≤Ai for each 1≤i≤N1≤i≤N
B1+B2+⋯+BN=MB1+B2+⋯+BN=M

Find the maximum value of gcd(A1−B1,A2−B2,…,AN−BN)gcd(A1−B1,A2−B2,…,AN−BN) over all good sequences BB. Here, gcdgcd denotes the greatest common divisor.

Note: gcd(a,b,c)=gcd(a,gcd(b,c))gcd(a,b,c)=gcd(a,gcd(b,c)) and gcd(a,0)=gcd(0,a)=agcd(a,0)=gcd(0,a)=a.

Input Format

The first line of input contains a single integer TT, denoting the number of test cases. The description of TT test cases follows.
The first line of each test case contains two space-separated integers N,MN,M.
The second line of each test case contains NN space-separated integers A1,A2,…,ANA1,A2,…,AN.

Output Format

For each test case, print a single line containing one integer — the maximum value of gcd(A1−B1,A2−B2,…,AN−BN)gcd(A1−B1,A2−B2,…,AN−BN) over all good sequences BB.

Constraints

1≤T≤101≤T≤10
1≤N≤1051≤N≤105
1≤Ai≤1051≤Ai≤105
0≤M<∑i=1NAi0≤M<∑i=1NAi

Subtasks

Subtask #1 (50 points): 1≤N≤1041≤N≤104

Subtask #2 (50 points): Original constraints

Sample Input 1

4
4 4
1 3 5 7
4 4
5 5 5 5
4 0
4 6 9 12
6 10
15 9 3 8 14 17

Sample Output 1

Explanation

Test case 11: The optimal strategy is to take B=(1,0,2,1)B=(1,0,2,1). The answer is gcd(1−1,3−0,5−2,7−1)=gcd(0,3,3,6)=3gcd(1−1,3−0,5−2,7−1)=gcd(0,3,3,6)=3.

Test case 22: The optimal strategy is to take B=(1,1,1,1)B=(1,1,1,1). The answer is gcd(5−1,5−1,5−1,5−1)=gcd(4,4,4,4)=4gcd(5−1,5−1,5−1,5−1)=gcd(4,4,4,4)=4.

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