102271: [AtCoder]ABC227 B - KEYENCE building
Description
Score : $200$ points
Problem Statement
There are $N$ people numbered $1$ to $N$.
Person $i$ guessed the building area of KEYENCE headquarters building to be $S_i$ square meters.
The shape of KEYENCE headquarters building is shown below, where $a$ and $b$ are some positive integers.
That is, the building area of the building can be represented as $4ab+3a+3b$.
Based on just this information, how many of the $N$ people are guaranteed to be wrong in their guesses?
Constraints
- $1 \leq N \leq 20$
- $1 \leq S_i \leq 1000$
- All values in input are integers.
Input
Input is given from Standard Input in the following format:
$N$ $S_1$ $\ldots$ $S_N$
Output
Print the answer.
Sample Input 1
3 10 20 39
Sample Output 1
1
The area would be $10$ square meters if $a=1,b=1$, and $39$ square meters if $a=2,b=3$.
However, no pair of positive integers $a$ and $b$ would make the area $20$ square meters.
Thus, we can only be sure that Person $2$ guessed wrong.
Sample Input 2
5 666 777 888 777 666
Sample Output 2
3
Input
题意翻译
有 $n$ 个正整数,记第 $i$ 个数为 $s_i$ 。求有多少个 $i$ 满足:不定方程 $4ab+3a+3b=s_i$ 无正整数解?Output
得分:$200$分
问题描述
有$N$个人,编号为$1$到$N$。
第$i$个人猜测KEYENCE总部大楼的建筑面积为$S_i$平方米。
KEYENCE总部大楼的形状如下图所示,其中$a$和$b$是一些正整数。
也就是说,该建筑的建筑面积可以用$4ab+3a+3b$来表示。
仅根据这些信息,有多少人肯定猜错了他们的猜测?
约束条件
- $1 \leq N \leq 20$
- $1 \leq S_i \leq 1000$
- 输入中的所有值都是整数。
输入
输入通过标准输入给出,格式如下:
$N$ $S_1$ $\ldots$ $S_N$
输出
输出答案。
样例输入1
3 10 20 39
样例输出1
1
如果$a=1,b=1$,建筑面积为$10$平方米;如果$a=2,b=3$,建筑面积为$39$平方米。
但是,没有任何一对正整数$a$和$b$可以使建筑面积为$20$平方米。
因此,我们只能确定第$2$个人猜错了。
样例输入2
5 666 777 888 777 666
样例输出2
3