USACO

USACO 2025 January Contest, Silver

Problem 2. Farmer John's Favorite Operation

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It is another cold and boring day on Farmer John's farm. To pass the time, Farmer John has invented a fun leisure activity involving performing operations on an integer array.

Farmer John has an array $a$ of $N$ ( $1 \leq N \leq 2 \cdot 10^5$ ) non-negative integers and an integer $M$ ( $1 \leq M \leq 10^9$ ). Then, FJ will ask Bessie for an integer $x$ . In one operation, FJ can pick an index $i$ and subtract or add $1$ to $a_i$ . FJ's boredom value is the minimum number of operations he must perform so that $a_i-x$ is divisible by $M$ for all $1 \leq i \leq N$ .

Among all possible $x$ , output FJ's minimum possible boredom value.

INPUT FORMAT (input arrives from the terminal / stdin):

The first line contains $T$ ( $1 \leq T \leq 10$ ), the number of independent test cases to solve.

The first line of each test case contains $N$ and $M$ .

The second line of each test case contains $a_1, a_2, ..., a_N$ ( $0 \leq a_i \leq 10^9$ ).

It is guaranteed that the sum of $N$ over all test cases does not exceed $5 \cdot 10^5$ .

OUTPUT FORMAT (print output to the terminal / stdout):

For each test case, output an integer on a new line containing FJ's minimum possible boredom value among all possible values of

$x$ .

SAMPLE INPUT:

2
5 9
15 12 18 3 8
3 69
1 988244353 998244853

SAMPLE OUTPUT:

10
21

In the first test case, one optimal choice of $x$ is $3$ . FJ can perform $10$ operations to make $a = [12, 12, 21, 3, 12]$ .

SCORING:

Input 2: $N \le 1000$ and $M \le 1000$ .
Input 3: $N\le 1000$ .
Inputs 4-5: $M\le 10^5$ .
Inputs 6-16: No additional constraints.

Problem credits: Chongtian Ma

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