Cyclic Sum

3000ms

524288K

Description:

Denote a cyclic sequence of size $$$n$$$ as an array $$$s$$$ such that $$$s_n$$$ is adjacent to $$$s_1$$$. The segment $$$s[r, l]$$$ where $$$l < r$$$ is the concatenation of $$$s[r, n]$$$ and $$$s[1, l]$$$.

You are given an array $$$a$$$ consisting of $$$n$$$ integers. Define $$$b$$$ as the cyclic sequence obtained from concatenating $$$m$$$ copies of $$$a$$$. Note that $$$b$$$ has size $$$n \cdot m$$$.

You are given an integer $$$k$$$ where $$$k = 1$$$ or $$$k$$$ is a prime number. Find the number of different segments in $$$b$$$ where the sum of elements in the segment is divisible by $$$k$$$.

Two segments are considered different if the set of indices of the segments are different. For example, when $$$n = 3$$$ and $$$m = 2$$$, the set of indices for segment $$$s[2, 5]$$$ is $$$\{2, 3, 4, 5\}$$$, and for segment $$$s[5, 2]$$$ is $$$\{5, 6, 1, 2\}$$$. In particular, the segments $$$s[1, 6], s[2,1], \ldots, s[6, 5]$$$ are considered as the same segment.

Output the answer modulo $$$10^9 + 7$$$.

Input:

The first line contains three integers $$$n$$$, $$$m$$$, and $$$k$$$ ($$$1 \leq n, m, k \leq 2 \cdot 10^5$$$, $$$k = 1$$$ or $$$k$$$ is a prime number).

The second line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$0 \leq a_i \leq 2 \cdot 10^5$$$).

Output:

Output an integer denoting the number of different segments in $$$b$$$ where the sum of elements in the segment is divisible by $$$k$$$, modulo $$$10^9 + 7$$$.

Sample Input:

5 1 5
1 2 3 4 3

Sample Output:

Sample Input:

5 1 5
1 2 3 4 5

Sample Output:

Sample Input:

5 4 5
1 2 3 4 5

Sample Output:

Note:

In the first example, all valid segments are $$$[1,4]$$$, $$$[2, 3]$$$, $$$[3, 5]$$$, and $$$[4, 2]$$$.

In the second example, one of the valid segments is $$$[1, 5]$$$.

Informação

Provedor Codeforces

Código CF1575C