[USACO16JAN] Subsequences Summing to Sevens S

题目描述

Farmer John’s $N$ cows are standing in a row, as they have a tendency to do from time to time. Each cow is labeled with a distinct integer ID number so FJ can tell them apart. FJ would like to take a photo of a contiguous group of cows but, due to a traumatic childhood incident involving the numbers $1\dots 6$, he only wants to take a picture of a group of cows if their IDs add up to a multiple of 7.

Please help FJ determine the size of the largest group he can photograph.

给你n个数，分别是a[1],a[2],…,a[n]。求一个最长的区间[x,y]，使得区间中的数(a[x],a[x+1],a[x+2],…,a[y-1],a[y])的和能被7整除。输出区间长度。若没有符合要求的区间，输出0。

输入格式

The first line of input contains $N$ ($1\le N\le 50,000$). The next $N$

lines each contain the $N$ integer IDs of the cows (all are in the range

$0\dots 1,000,000$).

输出格式

Please output the number of cows in the largest consecutive group whose IDs sum

to a multiple of 7. If no such group exists, output 0.

样例 #1

样例输入 #1

7
3
5
1
6
2
14
10

样例输出 #1

5

提示

In this example, 5+1+6+2+14 = 28.

题解

准备国庆假期了，脑子也变得不太好使，这种题我居然没有第一时间想到答案，非常难受。因为脑子进水了，所以我看了别人C++的题解，看到：

$(A-B)\mathrm{mod}num\equiv 0$
等价于
$A\equiv B\mathrm{mod}num$

唉，这个同余定理这么常用，为什么我时不时就把它给忘记呢？

大家可以去参考一下大佬的题解：题解 P3131 【[USACO16JAN]子共七Subsequences Summing to Sevens】

这里给出对应的Python代码：

def Solution2():
    N = int(input())
    Prefix = 0
    yvshu = {i: [] for i in range(7)}
    for i in range(N):
        Prefix += int(input())
        yvshu[Prefix%7].append(i+1)
    ans = max(yvshu[0]) if yvshu[0] else 0
    for key in yvshu.keys():
        if len(yvshu[key]) > 1:
            ans = max(ans, yvshu[key][-1] - yvshu[key][0])
    print(ans)

Solution2()

关于上面那个定理的题我还可以提供一题：
Ringo’s Favorite Numbers 2