数论-GCD、LCM、扩展欧几里得

最大公约数GCD

欧几里得算法（辗转相除法）求GCD

int gcd(int x, int y) {
    return y == 0 ? x : gcd(y, x % y);
}

最小公倍数LCM

int lcm(int x, int y) {
    return x / gcd(x, y) * y;
}

扩展欧几里得

问题引入：$ax+by=n$什么时候有整数解？有解的充要条件是$gcd(a,b)$可以整除$n$，当方程符合$ax+by=gcd(a,b)$时，可以用扩展欧几里得算法求一个整数解$(x_0,y_0)$，程序如下：

void extend_gcd(int a, int b, int& x, int& y) {   //返回x,y
    if (b == 0) {
        x = 1, y = 0;
        return;
    }
    extend_gcd(b, a % b, x, y);
    int temp = x;
    x = y;
    y = temp - (a / b) * y;
}

得到$(x_0,y_0)后，$进而推出方程$ax+by=n$的一个解$(x_0^{\prime },y_0^{\prime })，x_0^{\prime }=x_{0}n/gcd(a,b)，y_0^{\prime }=y_{0}n/gcd(a,b)$。

扩展欧几里得应用：

1. 求解不定方程
2. 求解摸的逆元
3. 求解同余方程

例题

HDU-5223

HDU-5223 GCD

Problem Description
In mathematics, the greatest common divisor (gcd) of two or more integers, when at least one of them is not zero, is the largest positive integer that divides the numbers without a remainder. For example, the GCD of 8 and 12 is 4.—Wikipedia
BrotherK and Ery like playing mathematic games. Today, they are playing a game with GCD.
BrotherK has an array A with N elements: A1 ~ AN, each element is a integer in [1, 10^9]. Ery has Q questions, the i-th question is to calculate
GCD(ALi, ALi+1, ALi+2, …, ARi), and BrotherK will tell her the answer.
BrotherK feels tired after he has answered Q questions, so Ery can only play with herself, but she don’t know any elements in array A. Fortunately, Ery remembered all her questions and BrotherK’s answer, now she wants to recovery the array A.
Input
The first line contains a single integer T, indicating the number of test cases.
Each test case begins with two integers N, Q, indicating the number of array A, and the number of Ery’s questions. Following Q lines, each line contains three integers Li, Ri and Ansi, describing the question and BrotherK’s answer.
T is about 10
2 ≤ N Q ≤ 1000
1 ≤ Li < Ri ≤ N
1 ≤ Ansi ≤ 109
Output
For each test, print one line.
If Ery can’t find any array satisfy all her question and BrotherK’s answer, print “Stupid BrotherK!” (without quotation marks). Otherwise, print N integer, i-th integer is Ai.
If there are many solutions, you should print the one with minimal sum of elements. If there are still many solutions, print any of them.
Sample Input
2
2 2
1 2 1
1 2 2
2 1
1 2 2
Sample Output
Stupid BrotherK!
2 2

给定若干区间的GCD，试还原原数组。
贪心乘最小的数使得区间内每个数是ans[i]的倍数(LCM)，最后再检查一遍。

#include
using namespace std;
typedef long long ll;
const int maxn = 1003;
ll t, n, q;
ll a[maxn], l[maxn], r[maxn], ans[maxn];
ll gcd(ll x, ll y) {
    return y == 0 ? x : gcd(y, x % y);
}
ll lcm(ll x, ll y) {
    return x / gcd(x, y) * y;
}
int main() {
    cin >> t;
    while (t--) {
        cin >> n >> q;
        for (int i = 1; i <= n; i++)a[i] = 1; //初始化原数组1
        for (int i = 1; i <= q; i++) {
            cin >> l[i] >> r[i] >> ans[i];
            for (int j = l[i]; j <= r[i]; j++)
                a[j] = lcm(a[j], ans[i]);
        }
        bool tag = true;
        for (int i = 1; i <= q; i++) {  //检验
        
            ll tmp = a[l[i]];
            for (ll j = l[i] + 1; j <= r[i]; j++)
                tmp = gcd(tmp, a[j]);
            if (tmp != ans[i]) {
                tag = false;
                break;
            }
        }
        if (tag) {
            cout << a[1];
            for (int i = 2; i <= n; i++)cout << " " << a[i];
            cout << "\n";
        }
        else cout << "Stupid BrotherK!\n";
    }
    return 0;
}

HDU-1576

HDU-1576 A/B

Problem Description
要求(A/B)%9973，但由于A很大，我们只给出n(n=A%9973)(我们给定的A必能被B整除，且gcd(B,9973) = 1)。
Input
数据的第一行是一个T，表示有T组数据。
每组数据有两个数n(0 <= n < 9973)和B(1 <= B <= 10^9)。
Output
对应每组数据输出(A/B)%9973。
Sample Input
2
1000 53
87 123456789
Sample Output
7922
6060

$令m=9973,x=a/b，则a=bx$
$又n=a\%m,所以n=a-a/m\ast m$
$bx代入a,n=bx-bx/m\ast m$
$设x_1=x,y_1=bx/m,则n=b{x}_1-y_{1}m$
$两边除n,\frac{b{x}_1}{n}-\frac{y_{1}m}{n}=1，即b{x}_2-m{y}_2=1$
$又因为gcd(b,m)=1,所以b{x}_2-m{y}_2=gcd(b,m)$
所以通过扩展欧几里得求得的最小特解$x_2$，$x_2$乘$n$得$x_1$，最后处理下取余负数即可。

#include
using namespace std;
#define mod 9973
int t, n, a, b, x, y;
void extend_gcd(int a, int b, int& x, int& y) {
    if (b == 0) {
        x = 1, y = 0;
        return;
    }
    extend_gcd(b, a % b, x, y);
    int temp = x;
    x = y;
    y = temp - (a / b) * y;
}
int main() {
    cin >> t;
    while (t--) {
        cin >> n >> b;
        extend_gcd(b, mod, x, y);
        x *= n;
        x = (x % mod + mod) % mod;
        cout << x << "\n";
    }
    return 0;
}

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