1902C · Insert and Equalize

Problem: 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 $x$?

Input Format: 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 $2 \cdot 10^5$.

Output Format: 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 $x$.

Note: 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 $a_{n+1}$.