P-2000 Problem: D2-2000 Problem

D2-2000 Problem

Question

Find the number of subsets of ,the sum of whose elements is divisible by .

对于集合，它有多少个子集满足这个子集中所有数之和能被整除?

Construction generating function

Factored

Expanded

Carry in special value

（ , or ）

P-2000 Problem

Find the number of subsets of {1,2,3,4,5,…,2000} , the sum of whose elements is prime.对于集合 {1,2,3,4,5,…,2000}，它有多少个子集满足这个子集中所有数之和是素数?

D5-2000 Problem

该文档探讨集合{1,2,...,2000}中元素和能被5整除的子集数量。通过递推法分组计算子集和得出递推公式，生成函数法则转化为多项式展开系数求和，利用单位根性质简化计算。最终求得结果为1/5(2^2000 + 4×2^400)，视频演示提供了直观理解。对于集合 {1,2,3,4,5,...,2000} ，它有多少个子集满足这个子集中所有数之和能被 5 整除?