1103 Integer Factorization

Statement

Metadata

作者: CHEN, Yue
单位: 浙江大学
代码长度限制: 16 KB
时间限制: 1200 ms
内存限制: 64 MB

The $K-P$ factorization of a positive integer $N$ is to write $N$ as the sum of the $P$ -th power of $K$ positive integers. You are supposed to write a program to find the $K-P$ factorization of $N$ for any positive integers $N$ , $K$ and $P$ .

Input Specification

Each input file contains one test case which gives in a line the three positive integers $N$ ( $\le 400$ ), $K$ ( $\le N$ ) and $P$ ( $1 < P\le 7$ ). The numbers in a line are separated by a space.

Output Specification

For each case, if the solution exists, output in the format:

N = n[1]^P + ... n[K]^P

where n[i] (i = 1, …, K) is the i-th factor. All the factors must be printed in non-increasing order.

Note: the solution may not be unique. For example, the 5-2 factorization of 169 has 9 solutions, such as $12^2 + 4^2 + 2^2 + 2^2 + 1^2$ , or $11^2 + 6^2 + 2^2 + 2^2 + 2^2$ , or more. You must output the one with the maximum sum of the factors. If there is a tie, the largest factor sequence must be chosen – sequence { $a_1, a_2, \cdots , a_K$ } is said to be larger than { $b_1, b_2, \cdots , b_K$ } if there exists $1\le L\le K$ such that $a_i=b_i$ for $i$ and $a_L>b_L$ .

If there is no solution, simple output Impossible.

Sample Input 1

169 5 2

Sample Output 1

169 = 6^2 + 6^2 + 6^2 + 6^2 + 5^2

Sample Input 2

169 167 3

Sample Output 2

Impossible

Last update: May 4, 2022