K-good

题意翻译

给定一个整数 $n$，请找出一个大于等于 $2$ 的整数 $k$，使得 $n$ 可以表示成 $k$ 个除以 $k$ 的余数互不相同的正整数之和。 数据范围： - $t$ 组数据，$1\leqslant t\leqslant 10^5$。 - $2\leqslant n\leqslant 10^{18}$。 Translated by Eason_AC

题目描述

We say that a positive integer $ n $ is $ k $ -good for some positive integer $ k $ if $ n $ can be expressed as a sum of $ k $ positive integers which give $ k $ distinct remainders when divided by $ k $ . Given a positive integer $ n $ , find some $ k \geq 2 $ so that $ n $ is $ k $ -good or tell that such a $ k $ does not exist.

输入输出格式

输入格式

The input consists of multiple test cases. The first line contains a single integer $ t $ ( $ 1 \leq t \leq 10^5 $ ) — the number of test cases. Each test case consists of one line with an integer $ n $ ( $ 2 \leq n \leq 10^{18} $ ).

输出格式

For each test case, print a line with a value of $ k $ such that $ n $ is $ k $ -good ( $ k \geq 2 $ ), or $ -1 $ if $ n $ is not $ k $ -good for any $ k $ . If there are multiple valid values of $ k $ , you can print any of them.

输入输出样例

输入样例 #1

5
2
4
6
15
20

输出样例 #1

-1
-1
3
3
5

说明

$ 6 $ is a $ 3 $ -good number since it can be expressed as a sum of $ 3 $ numbers which give different remainders when divided by $ 3 $ : $ 6 = 1 + 2 + 3 $ . $ 15 $ is also a $ 3 $ -good number since $ 15 = 1 + 5 + 9 $ and $ 1, 5, 9 $ give different remainders when divided by $ 3 $ . $ 20 $ is a $ 5 $ -good number since $ 20 = 2 + 3 + 4 + 5 + 6 $ and $ 2,3,4,5,6 $ give different remainders when divided by $ 5 $ .