A15374. Divisibility by 2^n
编程题
普及/提高-
知识点
题目描述
You are given an array of positive integers $a_1, a_2, \ldots, a_n$ .
Make the product of all the numbers in the array (that is, $a_1 \cdot a_2 \cdot \ldots \cdot a_n$ ) divisible by $2^n$ .
You can perform the following operation as many times as you like:
- select an arbitrary index $i$ ( $1 \leq i \leq n$ ) and replace the value $a_i$ with $a_i=a_i \cdot i$ .
You cannot apply the operation repeatedly to a single index. In other words, all selected values of $i$ must be different.
Find the smallest number of operations you need to perform to make the product of all the elements in the array divisible by $2^n$ . Note that such a set of operations does not always exist.
Make the product of all the numbers in the array (that is, $a_1 \cdot a_2 \cdot \ldots \cdot a_n$ ) divisible by $2^n$ .
You can perform the following operation as many times as you like:
- select an arbitrary index $i$ ( $1 \leq i \leq n$ ) and replace the value $a_i$ with $a_i=a_i \cdot i$ .
You cannot apply the operation repeatedly to a single index. In other words, all selected values of $i$ must be different.
Find the smallest number of operations you need to perform to make the product of all the elements in the array divisible by $2^n$ . Note that such a set of operations does not always exist.
输入格式
The first line of the input contains a single integer $t$ $(1 \leq t \leq 10^4$ ) — the number test cases.
Then the descriptions of the input data sets follow.
The first line of each test case contains a single integer $n$ ( $1 \leq n \leq 2 \cdot 10^5$ ) — the length of array $a$ .
The second line of each test case contains exactly $n$ integers: $a_1, a_2, \ldots, a_n$ ( $1 \leq a_i \leq 10^9$ ).
It is guaranteed that the sum of $n$ values over all test cases in a test does not exceed $2 \cdot 10^5$ .
Then the descriptions of the input data sets follow.
The first line of each test case contains a single integer $n$ ( $1 \leq n \leq 2 \cdot 10^5$ ) — the length of array $a$ .
The second line of each test case contains exactly $n$ integers: $a_1, a_2, \ldots, a_n$ ( $1 \leq a_i \leq 10^9$ ).
It is guaranteed that the sum of $n$ values over all test cases in a test does not exceed $2 \cdot 10^5$ .
输出格式
For each test case, print the least number of operations to make the product of all numbers in the array divisible by $2^n$ . If the answer does not exist, print -1.
输入输出样例
输入 #1
6 1 2 2 3 2 3 10 6 11 4 13 17 1 1 5 1 1 12 1 1 6 20 7 14 18 3 5
输出 #1
0 1 1 -1 2 1
说明/提示
In the first test case, the product of all elements is initially $2$ , so no operations needed.
In the second test case, the product of elements initially equals $6$ . We can apply the operation for $i = 2$ , and then $a_2$ becomes $2\cdot2=4$ , and the product of numbers becomes $3\cdot4=12$ , and this product of numbers is divided by $2^n=2^2=4$ .
In the fourth test case, even if we apply all possible operations, we still cannot make the product of numbers divisible by $2^n$ — it will be $(13\cdot1)\cdot(17\cdot2)\cdot(1\cdot3)\cdot(1\cdot4)=5304$ , which does not divide by $2^n=2^4=16$ .
In the fifth test case, we can apply operations for $i = 2$ and $i = 4$ .
In the second test case, the product of elements initially equals $6$ . We can apply the operation for $i = 2$ , and then $a_2$ becomes $2\cdot2=4$ , and the product of numbers becomes $3\cdot4=12$ , and this product of numbers is divided by $2^n=2^2=4$ .
In the fourth test case, even if we apply all possible operations, we still cannot make the product of numbers divisible by $2^n$ — it will be $(13\cdot1)\cdot(17\cdot2)\cdot(1\cdot3)\cdot(1\cdot4)=5304$ , which does not divide by $2^n=2^4=16$ .
In the fifth test case, we can apply operations for $i = 2$ and $i = 4$ .