A16292. Insert and Equalize

You are given an integer array $a_1, a_2, \dots, a_n$ , all its elements are distinct.

First, you are asked to insert one more integer $a_{n+1}$ into this array. $a_{n+1}$ should not be equal to any of $a_1, a_2, \dots, a_n$ .

Then, you will have to make all elements of the array equal. At the start, you choose a positive integer $x$ ( $x > 0$ ). In one operation, you add $x$ to exactly one element of the array. Note that $x$ is the same for all operations.

What's the smallest number of operations it can take you to make all elements equal, after you choose $a_{n+1}$ and $x$ ?

The first line contains a single integer $t$ ( $1 \le t \le 10^4$ ) — the number of testcases.

The first line of each testcase contains a single integer $n$ ( $1 \le n \le 2 \cdot 10^5$ ).

The second line contains $n$ integers $a_1, a_2, \dots, a_n$ ( $-10^9 \le a_i \le 10^9$ ). All $a_i$ are distinct.

The sum of $n$ over all testcases doesn't exceed $2 \cdot 10^5$ .

For each testcase, print a single integer — the smallest number of operations it can take you to make all elements equal, after you choose integers $a_{n+1}$ and $x$ .

In the first testcase, you can choose $a_{n+1} = 4$ , the array becomes $[1, 2, 3, 4]$ . Then choose $x = 1$ and apply the operation $3$ times to the first element, $2$ times to the second element, $1$ time to the third element and $0$ times to the fourth element.

In the second testcase, you can choose $a_{n+1} = 13, x = 4$ .

In the third testcase, you can choose $a_{n+1} = 9, x = 1$ . Then apply the operation once to $a_{n+1}$ .