# [COCI2017-2018#5] Pictionary

## 题意翻译

## 题目描述
在宇宙一个不为人知的地方，有一个星球，上面有一个国家，只有数学家居住。
在这个国家有$n$个数学家，有趣的是，每个数学家都住在自己的城市，且城市间无道路相连，因为他们可以在线交流。当然，城市有从$1$到$n$的编号。
一位数学家决定用手机发论文，而手机将“不言而喻”自动更正成了“猜谜游戏”。
不久之后，这个国家就发现了猜谜游戏。他们想要见面一起玩，于是这个国家就开始了修路工程。
道路修建会持续$m$天。对于第$i$天，若$\gcd(a,b)=m-i+1$，则$a$和$b$城市间会修一条路。
由于数学家们忙于建筑工作，请你来确定一对数学家最早什么时候能凑到一起玩。
## 输入输出格式
### 输入格式
第一行有三个正整数$n,m,q$，表示城市数量、修路持续天数、询问数量。
接下来$q$行，每行有两个正整数$a,b$，表示询问$a$和$b$两个城市的数学家最早什么时候能在一起玩。
### 输出格式
输出$q$行，第$i$行有一个正整数，表示第$i$次询问的结果
## 说明
**数据范围：**
对于$40\%$的数据：
$n≤4000,q≤10^5$
对于全部数据：
$1≤n,q≤10^5$
$1≤m≤n$
样例1解释：
在第一天，$(3,6)$之间修了一条路，因此第二次询问输出`1`
在第二天，$(2,4),(2,6),(2,8),(4,6),(6,8)$之间都修了一条路，此时$4$和$8$号城市连通，第三次询问输出`2`
在第三天，所有编号互质的城市之间都修了路，$2$和$5$号城市在此时连通，第一次询问输出`1`
样例2解释：
在第二天，$(20,15)$之间修了一条路
第四天，$(15,9)$之间修了一条路
所以$20$和$9$号城市在第四天连通，输出`4`

## 题目描述

There is a planet, in a yet undiscovered part of the universe, with a country inhabited solely
by mathematicians. In this country, there are a total of N mathematicians, and the interesting
fact is that each mathematician lives in their own city. Is it also interesting that no two cities
are connected with a road, because mathematicians can communicate online or by
reviewing academic papers. Naturally, the cities are labeled with numbers from 1 to N.
Life was perfect until one mathematician decided to write an academic paper on their
smartphone. The smartphone auto-corrected the word “self-evident” to “Pictionary” and the
paper was published as such. Soon after, the entire country discovered pictionary and
wanted to meet up and play, so construction work on roads between cities began shortly.
.
The road construction will last a total of M days, according to the following schedule: on the
first day, construction is done on roads between all pairs of cities that have M as their
greatest common divisor. On the second day, construction is done on roads between all
pairs of cities that have M-1 as their greatest common divisor, and so on until the $M^{th}$ day
when construction is done on roads between all pairs of cities that are co-prime. More
formally, on the $i^{th}$ day, construction is done on roads between cities a and b if gcd(a, b) = $M-i+1$.
Since the mathematicians are busy with construction work, they’ve asked you to help them
determine the minimal number of days before a given pair of mathematicians can play
pictionary together.

## 输入输出格式

### 输入格式

The first line of input contains three positive integers N, M
and Q
(1 ≤ N
, Q ≤ 100 000, 1 ≤ M
≤ N
), the number of cities, the number of days it takes to build the roads, and the number of
queries.
Each of the following Q lines contains two distinct positive integers A and B
(1 ≤ A
, B ≤ N
)
that denote the cities of the mathematicians who want to find out the minimal number of days
before they can play pictionary together.

### 输出格式

The $i^{th}$ line must contain the minimal number of days before the mathematicians from the $i^{th}$ query can play pictionary together.

## 输入输出样例

### 输入样例 #1

```
8 3 3
2 5
3 6
4 8
```

### 输出样例 #1

```
3
1
2
```

### 输入样例 #2

```
25 6 1
20 9
```

### 输出样例 #2

`4`

### 输入样例 #3

```
9999 2222 2
1025 2405
3154 8949
```

### 输出样例 #3

```
1980
2160
```

## 说明

In test cases worth 40% of total points, it will hold N
≤ 1000, Q
≤ 100 000.
**Clarification of the first test case:**
On the first day, road (3, 6) is built. Therefore the answer to the second query is 1.
On the second day, roads (2, 4), (2, 6), (2, 8), (4, 6) and (6, 8) are built. Cities 4 and 8 are now
connected (it is possible to get from the first to the second using city 6).
On the third day, roads between relatively prime cities are built, so cities 2 and 5 are connected.
**Clarification of the second test case:**
On the second day, road (20, 15) is built, whereas on the fourth day, road (15, 9) is built. After the
fourth day, cities 20 and 9 are connected via city 15.