D - Candy Distribution

Score : $400$ points

Problem Statement

There are $N$ boxes arranged in a row from left to right. The $i$-th box from the left contains $A_i$ candies.

You will take out the candies from some consecutive boxes and distribute them evenly to $M$ children.

Such being the case, find the number of the pairs $(l, r)$ that satisfy the following:

• $l$ and $r$ are both integers and satisfy $1 \leq l \leq r \leq N$.
• $A_l + A_{l+1} + ... + A_r$ is a multiple of $M$.

Constraints

• All values in input are integers.
• $1 \leq N \leq 10^5$
• $2 \leq M \leq 10^9$
• $1 \leq A_i \leq 10^9$

Input

Input is given from Standard Input in the following format:

$N$ $M$

$A_1$ $A_2$ $...$ $A_N$

Output

Print the number of the pairs $(l, r)$ that satisfy the conditions.

Note that the number may not fit into a $32$-bit integer type.

3 2
4 1 5

3

The sum $A_l + A_{l+1} + ... + A_r$ for each pair $(l, r)$ is as follows:

• Sum for $(1, 1)$: $4$
• Sum for $(1, 2)$: $5$
• Sum for $(1, 3)$: $10$
• Sum for $(2, 2)$: $1$
• Sum for $(2, 3)$: $6$
• Sum for $(3, 3)$: $5$

Among these, three are multiples of $2$.

13 17
29 7 5 7 9 51 7 13 8 55 42 9 81

6

10 400000000
1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000

25