The document describes an algorithm to solve a problem on an online judge platform. The problem involves counting the number of subarrays of integers whose sum is divisible by a given number M. The algorithm works by calculating a running sum of the integers, and checking if the remainder of the running sum modulo M is the same for any two indices, which would indicate the subarray between them has a sum divisible by M. It runs in O(N) time where N is the number of integers.

1. choi@startlink.io

2. https://www.acmicpc.net/problem/10986
• N
A1,
A2,
…,
AN
•
M
• ,
Ai
+
…
+
Aj
M
(i,
j)
• 1
≤
N
≤
106
• 2
≤
M
≤
103
• 0
≤
Ai
≤
109

3. https://www.acmicpc.net/problem/10986
•
N^2
• O(N^2)
=
(10^6)^2
=
10^12
• 10^12
≥
1
• !

4. https://www.acmicpc.net/problem/10986
• Si
=
A1
+
…
+
Ai
• Ai +
…
+
Aj =
Sj – Si-­‐1
• (Ai +
…
+
Aj)
%
M
=
(Sj – Si-­‐1)
%
M
• (Ai +
…
+
Aj)
%
M
==
0
• (Sj – Si-­‐1)
%
M ==
0
•
0
• Sj
%
M
==
Si-­‐1
%
M

5. https://www.acmicpc.net/problem/10986
•
• Sj %
M
==
Si-­‐1 %
M
•
(i,
j)
.
• cntk
Si
%
M
==
k
i
• 0
≤
k
<
M
k
• cntk
*
(cntk
-­‐
1)
/
2
.

6. https://www.acmicpc.net/problem/10986
• https://gist.github.com/Baekjoon/6d45e5ef5d7eed6039b5