【算法笔记自学】第 5 章入门篇（3）—

5.1简单数学

#include <cstdio>
#include <algorithm>
using namespace std;
bool cmp(int a,int b){return a>b;
}
void to_array(int n,int num[]){for(int i=0;i<4;i++){num[i]=n%10;n /=10;}
}
int to_number(int num[]){int sum=0;for(int i=0;i<4;i++){sum=sum*10+num[i];}return sum;
}
int main(){int n,MIN,MAX;scanf("%d",&n);int num[5];while(1){to_array(n,num);sort(num,num+4);MIN=to_number(num);sort(num,num+4,cmp);MAX=to_number(num);n=MAX-MIN;printf("%04d-%04d=%04d\n",MAX,MIN,n);if(n==0||n==6174)break;}return 0;
}

#include <cstdio>
#include <cmath>int main() {int a, b, c;scanf("%d%d%d", &a, &b, &c);int delta = b * b - 4 * a * c;if (delta < 0) {printf("No Solution");} else if (delta == 0) {printf("%.2f", -b / (2.0 * a));} else {printf("%.2f %.2f",  (-b - sqrt((double)delta)) / (2.0 * a), (-b + sqrt((double)delta)) / (2.0 * a));}return 0;
}

5.2最大公约数与最小公倍数

#include <cstdio>
#include <cmath>
int gcd(int a,int b){if(b==0)return a;//求最大公约数的辗转相除法递归写法else return gcd(b,a%b);
}
int main() {int m,n;while(scanf("%d%d",&m,&n)!=EOF){printf("%d\n",gcd(m,n));}return 0;
}

#include <cstdio>int gcd(int a, int b) {if (b == 0) {return a;} else {return gcd(b, a % b);}
}int main() {int a, b;scanf("%d%d", &a, &b);printf("%d", a / gcd(a, b) * b);return 0;
}

5.3分数的四则运算

#include <cstdio>
#include <algorithm>
using namespace std;struct Fraction {int up, down;
};int gcd(int a, int b) {if (b == 0) {return a;} else {return gcd(b, a % b);}
}Fraction reduction(Fraction fraction) {if (fraction.down < 0) {fraction.up = -fraction.up;fraction.down = -fraction.down;}if (fraction.up == 0) {fraction.down = 1;} else {int d = gcd(abs(fraction.up), abs(fraction.down));fraction.up /= d;fraction.down /= d;}return fraction;
}int main() {Fraction fraction;scanf("%d%d", &fraction.up, &fraction.down);Fraction result = reduction(fraction);if (result.down == 1) {printf("%d", result.up);} else {printf("%d %d", result.up, result.down);}return 0;
}

#include <cstdio>
#include <algorithm>
using namespace std;struct Fraction {int up, down;
};int gcd(int a, int b) {if (b == 0) {return a;} else {return gcd(b, a % b);}
}Fraction reduction(Fraction fraction) {if (fraction.down < 0) {fraction.up = -fraction.up;fraction.down = -fraction.down;}if (fraction.up == 0) {fraction.down = 1;} else {int d = gcd(abs(fraction.up), abs(fraction.down));fraction.up /= d;fraction.down /= d;}return fraction;
}Fraction add(Fraction f1, Fraction f2) {Fraction result;result.up = f1.up * f2.down + f2.up * f1.down;result.down = f1.down * f2.down;return reduction(result);
}int main() {Fraction f1, f2;scanf("%d%d%d%d", &f1.up, &f1.down, &f2.up, &f2.down);Fraction result = add(f1, f2);if (result.down == 1) {printf("%d", result.up);} else {printf("%d %d", result.up, result.down);}return 0;
}

#include <cstdio>
#include <algorithm>
using namespace std;struct Fraction {int up, down;
};int gcd(int a, int b) {if (b == 0) {return a;} else {return gcd(b, a % b);}
}Fraction reduction(Fraction fraction) {if (fraction.down < 0) {fraction.up = -fraction.up;fraction.down = -fraction.down;}if (fraction.up == 0) {fraction.down = 1;} else {int d = gcd(abs(fraction.up), abs(fraction.down));fraction.up /= d;fraction.down /= d;}return fraction;
}Fraction sub(Fraction f1, Fraction f2) {Fraction result;result.up = f1.up * f2.down - f2.up * f1.down;result.down = f1.down * f2.down;return reduction(result);
}int main() {Fraction f1, f2;scanf("%d%d%d%d", &f1.up, &f1.down, &f2.up, &f2.down);Fraction result = sub(f1, f2);if (result.down == 1) {printf("%d", result.up);} else {printf("%d %d", result.up, result.down);}return 0;
}

#include <cstdio>
#include <algorithm>
using namespace std;struct Fraction {int up, down;
};int gcd(int a, int b) {if (b == 0) {return a;} else {return gcd(b, a % b);}
}Fraction reduction(Fraction fraction) {if (fraction.down < 0) {fraction.up = -fraction.up;fraction.down = -fraction.down;}if (fraction.up == 0) {fraction.down = 1;} else {int d = gcd(abs(fraction.up), abs(fraction.down));fraction.up /= d;fraction.down /= d;}return fraction;
}Fraction multiply(Fraction f1, Fraction f2) {Fraction result;result.up = f1.up * f2.up;result.down = f1.down * f2.down;return reduction(result);
}int main() {Fraction f1, f2;scanf("%d%d%d%d", &f1.up, &f1.down, &f2.up, &f2.down);Fraction result = multiply(f1, f2);if (result.down == 1) {printf("%d", result.up);} else {printf("%d %d", result.up, result.down);}return 0;
}

#include <cstdio>
#include <algorithm>
using namespace std;struct Fraction {int up, down;
};int gcd(int a, int b) {if (b == 0) {return a;} else {return gcd(b, a % b);}
}Fraction reduction(Fraction fraction) {if (fraction.down < 0) {fraction.up = -fraction.up;fraction.down = -fraction.down;}if (fraction.up == 0) {fraction.down = 1;} else {int d = gcd(abs(fraction.up), abs(fraction.down));fraction.up /= d;fraction.down /= d;}return fraction;
}Fraction div(Fraction f1, Fraction f2) {Fraction result;result.up = f1.up * f2.down;result.down = f1.down * f2.up;return reduction(result);
}int main() {Fraction f1, f2;scanf("%d%d%d%d", &f1.up, &f1.down, &f2.up, &f2.down);Fraction result = div(f1, f2);if(!f2.up){printf("undefined");}else if (result.down == 1) {printf("%d", result.up);} else {printf("%d %d", result.up, result.down);}return 0;
}

5.4素数

#include <cstdio>
#include <algorithm>
#include <cmath>
using namespace std;
bool isPrime(int n){if(n<=1)return false;int sqr=(int)sqrt(1.0*n);for(int i=2;i<=sqr;i++){if(n%i==0)return false;}return true;}
int main() {int n;scanf("%d",&n);if(isPrime(n))printf("Yes");else printf("No");return 0;
}

#include <cstdio>
#include <algorithm>
#include <cmath>
using namespace std;
bool isPrime(int n){if(n<=1)return false;int sqr=(int)sqrt(1.0*n);for(int i=2;i<=sqr;i++){if(n%i==0)return false;}return true;}
int main() {int n;scanf("%d",&n);for(int i=1;i<n+1;i++){if(isPrime(i))printf("%d\n",i);}return 0;
}

5.5质因子分解

#include <cstdio>
int main() {int n;scanf("%d", &n);int counter = 0;while (n % 2 == 0) {counter++;n /= 2;}printf("%d", counter);return 0;
}

#include <cstdio>
#include <cmath>
#include <cstring>
#include <vector>
using namespace std;const int MAXN = 1000 + 1;
bool isPrime[MAXN];
vector<int> primes;void getPrimes(int n) {memset(isPrime, true, sizeof(isPrime));for (int i = 2; i <= n; i++) {if (isPrime[i]) {primes.push_back(i);for (int j = i + i; j <= n; j += i) {isPrime[j] = false;}}}
}int main() {int n;scanf("%d", &n);getPrimes((int)sqrt(1.0 * n));for (int i = 0; i < primes.size() && n > 1; i++) {int counter = 0;while (n > 1 && n % primes[i] == 0) {counter++;n /= primes[i];}if (counter > 0) {printf("%d %d\n", primes[i], counter);}}if (n > 1) {printf("%d 1", n);}return 0;
}

5.6大整数运算

#include <iostream>
#include <vector>
#include <string>
#include <algorithm>
using namespace std;typedef vector<int> BigInt;BigInt toBigInt(string nums) {BigInt result;for (int i = (int)nums.length() - 1; i >= 0; i--) {result.push_back(nums[i] - '0');}return result;
}int compare(BigInt a, BigInt b) {if (a.size() > b.size()) {return 1;} else if (a.size() < b.size()) {return -1;} else {for (int i = (int)a.size() - 1; i >= 0; i--) {if (a[i] > b[i]) {return 1;} else if (a[i] < b[i]) {return -1;}}return 0;}
}int main() {string nums1, nums2;cin >> nums1 >> nums2;BigInt a = toBigInt(nums1);BigInt b = toBigInt(nums2);int compareResult = compare(a, b);if (compareResult < 0) {printf("a < b");} else if (compareResult > 0) {printf("a > b");} else {printf("a = b");}
}

#include <iostream>
#include <vector>
#include <string>
#include <algorithm>
using namespace std;typedef vector<int> BigInt;BigInt toBigInt(string nums) {BigInt result;for (int i = (int)nums.length() - 1; i >= 0; i--) {result.push_back(nums[i] - '0');}return result;
}BigInt add(BigInt a, BigInt b) {BigInt c;int carry = 0;for (int i = 0; i < a.size() || i < b.size(); i++) {int aDigit = i < a.size() ? a[i] : 0;int bDigit = i < b.size() ? b[i] : 0;int sum = aDigit + bDigit + carry;c.push_back(sum % 10);carry = sum / 10;}if (carry) {c.push_back(carry);}return c;
}void print(BigInt a) {for (int i = (int)a.size() - 1; i >= 0; i--) {cout << a[i];}
}int main() {string nums1, nums2;cin >> nums1 >> nums2;BigInt a = toBigInt(nums1);BigInt b = toBigInt(nums2);print(add(a, b));return 0;
}

#include <iostream>
#include <vector>
#include <string>
#include <algorithm>
using namespace std;typedef vector<int> BigInt;BigInt toBigInt(string nums) {BigInt result;for (int i = (int)nums.length() - 1; i >= 0; i--) {result.push_back(nums[i] - '0');}return result;
}
int compare(BigInt a, BigInt b) {if (a.size() > b.size()) {return 1;} else if (a.size() < b.size()) {return -1;} else {for (int i = (int)a.size() - 1; i >= 0; i--) {if (a[i] > b[i]) {return 1;} else if (a[i] < b[i]) {return -1;}}return 0;}
}
BigInt sub(BigInt a, BigInt b) {BigInt c;for (int i = 0; i < a.size() || i < b.size(); i++) {int bDigit = i < b.size() ? b[i] : 0;if (a[i] < bDigit) {a[i + 1]--;a[i] += 10;}c.push_back(a[i] - bDigit);}while (c.size() > 1 && c.back() == 0) {c.pop_back();}return c;
}void print(BigInt a) {for (int i = (int)a.size() - 1; i >= 0; i--) {cout << a[i];}
}int main() {string nums1, nums2;cin >> nums1 >> nums2;BigInt a = toBigInt(nums1);BigInt b = toBigInt(nums2);if (compare(a, b) >= 0) {print(sub(a, b));} else {cout << "-";print(sub(b, a));}return 0;
}

#include <iostream>
#include <vector>
#include <string>
#include <algorithm>
using namespace std;typedef vector<int> BigInt;BigInt toBigInt(string nums) {BigInt result;for (int i = (int)nums.length() - 1; i >= 0; i--) {result.push_back(nums[i] - '0');}return result;
}
int compare(BigInt a, BigInt b) {if (a.size() > b.size()) {return 1;} else if (a.size() < b.size()) {return -1;} else {for (int i = (int)a.size() - 1; i >= 0; i--) {if (a[i] > b[i]) {return 1;} else if (a[i] < b[i]) {return -1;}}return 0;}
}
BigInt mul(BigInt a, int b) {BigInt c;int carry=0;;for (int i = 0; i < a.size(); i++) {int temp=a[i]*b+carry;c.push_back(temp%10);carry=temp/10;}while(carry!=0){c.push_back(carry%10);carry/=10;}while (c.size() > 1 && c.back() == 0) {c.pop_back();}return c;
}void print(BigInt a) {for (int i = (int)a.size() - 1; i >= 0; i--) {cout << a[i];}
}int main() {string nums;int b;cin >> nums >> b;BigInt a = toBigInt(nums);print(mul(a, b));return 0;return 0;
}

#include <iostream>
#include <vector>
#include <string>
#include <algorithm>
using namespace std;typedef vector<int> BigInt;BigInt toBigInt(string nums) {BigInt result;for (int i = (int)nums.length() - 1; i >= 0; i--) {result.push_back(nums[i] - '0');}return result;
}BigInt mul(BigInt a, BigInt b) {BigInt c = BigInt(a.size() + b.size() + 1, 0);for (int i = 0; i < a.size(); i++) {for (int j = 0; j < b.size(); j++) {c[i + j] += a[i] * b[j];}}for (int i = 0; i < a.size() + b.size(); i++) {if (c[i] >= 10) {c[i + 1] += c[i] / 10;c[i] = c[i] % 10;}}while (c.size() > 1 && c.back() == 0) {c.pop_back();}return c;
}void print(BigInt a) {for (int i = (int)a.size() - 1; i >= 0; i--) {cout << a[i];}
}int main() {string nums1, nums2;cin >> nums1 >> nums2;BigInt a = toBigInt(nums1);BigInt b = toBigInt(nums2);print(mul(a, b));return 0;
}

#include <iostream>
#include <vector>
#include <string>
#include <algorithm>
using namespace std;typedef vector<int> BigInt;BigInt toBigInt(string nums) {BigInt result;for (int i = (int)nums.length() - 1; i >= 0; i--) {result.push_back(nums[i] - '0');}return result;
}BigInt div(BigInt a, int b, int &r) {BigInt c;for (int i = (int)a.size() - 1; i >= 0; i--) {r = r * 10 + a[i];c.push_back(r / b);r = r % b;}reverse(c.begin(), c.end());while (c.size() > 1 && c.back() == 0) {c.pop_back();}return c;
}void print(BigInt a) {for (int i = (int)a.size() - 1; i >= 0; i--) {cout << a[i];}
}int main() {string nums;int b, r = 0;cin >> nums >> b;if (b == 0) {cout << "undefined";return 0;}BigInt a = toBigInt(nums);BigInt q = div(a, b, r);print(q);cout << " " << r;return 0;
}

5.7扩展欧几里得算法

#include <cstdio>
#include <algorithm>
using namespace std;int gcd(int a, int b) {if (b == 0) {return a;} else {return gcd(b, a % b);}
}int main() {int a, b, c;scanf("%d%d%d", &a, &b, &c);printf(c % gcd(a, b) == 0 ? "Yes" : "No");return 0;
}

#include <cstdio>
#include <algorithm>
using namespace std;int exGcd(int a, int b, int &x, int &y) {if (b == 0) {x = 1;y = 0;return a;}int d = exGcd(b, a % b, x, y);int temp = x;x = y;y = temp - a / b * y;return d;
}int main() {int a, b, x, y;scanf("%d%d", &a, &b);int d = exGcd(a, b, x, y);int step = b / d;int minX = (x % step + step) % step;printf("%d %d", minX, (d - a * minX) / b);return 0;
}

#include <cstdio>
#include <algorithm>
using namespace std;int exGcd(int a, int b, int &x, int &y) {if (b == 0) {x = 1;y = 0;return a;}int d = exGcd(b, a % b, x, y);int temp = x;x = y;y = temp - a / b * y;return d;
}int solve(int a, int b, int c) {int x, y;int d = exGcd(a, b, x, y);if (c % d) {return -1;} else {int step = abs(b / d);int minX = (c * x / d % step + step) % step;return minX;}
}int main() {int a, b, c;scanf("%d%d%d", &a, &b, &c);int minX = solve(a, b, c);if (minX == -1) {printf("No Solution");} else {printf("%d %d", minX, (c - a * minX) / b);}return 0;
}

#include <cstdio>
#include <algorithm>
using namespace std;int exGcd(int a, int b, int &x, int &y) {if (b == 0) {x = 1;y = 0;return a;}int d = exGcd(b, a % b, x, y);int temp = x;x = y;y = temp - a / b * y;return d;
}int solve(int a, int b, int c) {int x, y;int d = exGcd(a, b, x, y);if (c % d) {return -1;} else {int step = abs(b / d);int minX = (c * x / d % step + step) % step;return minX;}
}int main() {int a, c, m, x, y;scanf("%d%d%d", &a, &c, &m);int minX = solve(a, m, c);if (minX == -1) {printf("No Solution");} else {printf("%d", minX);}return 0;
}

#include <cstdio>
#include <algorithm>
using namespace std;int exGcd(int a, int b, int &x, int &y) {if (b == 0) {x = 1;y = 0;return a;}int d = exGcd(b, a % b, x, y);int temp = x;x = y;y = temp - a / b * y;return d;
}int invert(int a, int m) {int x, y;int d = exGcd(a, m, x, y);if (d != 1) {return -1;} else {return (x % m + m) % m;}
}int main() {int a, m;scanf("%d%d", &a, &m);int result = invert(a, m);if (result == -1) {printf("No Solution");} else {printf("%d", result);}return 0;
}

#include <cstdio>
#include <algorithm>
using namespace std;int exGcd(int a, int b, int &x, int &y) {if (b == 0) {x = 1;y = 0;return a;}int d = exGcd(b, a % b, x, y);int temp = x;x = y;y = temp - a / b * y;return d;
}int invert(int a, int m) {int x, y;int d = exGcd(a, m, x, y);if (d != 1) {return -1;} else {return (x % m + m) % m;}
}int main() {int n, a, m, b;scanf("%d%d%d", &n, &a, &m);int result = invert(abs(a), m);for (int i = 0; i < n; i++) {scanf("%d", &b);result = (result * b) % m;}printf("%d", result);return 0;
}

5.8组合数

#include <cstdio>
int cal(int n,int p)
{if(n<p)return 0;return n/p+cal(n/p,p);}
int main() {int n,p=2;scanf("%d", &n);printf("%d", cal(n,p));return 0;
}

#include <cstdio>typedef long long LL;LL C(LL n, LL m) {LL ans = 1;for (LL i = 1; i <= m; i++) {ans = ans * (n - m + i) / i;}return ans;
}int main() {LL n, m;scanf("%lld%lld", &n, &m);printf("%lld", C(n, m));return 0;
}