数论基础

扩展欧几里得

hdu2669, 解同余方程 ax+by=1

 1 #include <iostream>

 2 #include <cstdio>

 3 #include <fstream>

 4 #include <algorithm>

 5 #include <cmath>

 6 #include <deque>

 7 #include <vector>

 8 #include <queue>

 9 #include <string>

10 #include <cstring>

11 #include <map>

12 #include <stack>

13 #include <set>

14 //#define OPEN_FILE

15 #define INF 0x3f3f3f3f

16 using namespace std;

17 int x, y, a, b;

18 int gcd(int a, int b, int& x, int& y)

19 {

20     if(b == 0){

21         x = 1;

22         y = 0;

23         return a;

24     }

25     else{

26         int temp = gcd(b, a % b, x, y);

27         int t = y;

28         y = x - y * (a / b);

29         x = t;

30         return temp;

31     }

32 }

33 

34 int main()

35 {

36 #ifdef OPEN_FILE

37     freopen("in.txt", "r", stdin);

38     freopen("out.txt", "w", stdout);

39 #endif // OPEN_FILE

40     while(~scanf("%d%d", &a, &b)){

41         if(gcd(a, b ,x ,y) > 1){

42             printf("sorry\n");

43             continue;

44         }

45         while(x < 0){

46             x += b;

47             y -= a;

48         }

49         printf("%d %d\n", x, y);

50     }

51 

52 }

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费马小定理 gcd(a,p)=1，那么 a^(p-1) ≡1（mod p）

hdu4828,费马小定理计算卡特兰数

 1 #include <iostream>

 2 #include <cstdio>

 3 #include <fstream>

 4 #include <algorithm>

 5 #include <cmath>

 6 #include <deque>

 7 #include <vector>

 8 #include <queue>

 9 #include <string>

10 #include <cstring>

11 #include <map>

12 #include <stack>

13 #include <set>

14 //#define OPEN_FILE

15 #define INF 0x3f3f3f3f

16 #define MAXN 2000001

17 #define MOD 1000000007

18 using namespace std;

19 

20 long long f[MAXN + 5];

21 long long x, y;

22 long long gcd(long long a, long long b, long long& x, long long& y)

23 {

24     if(b == 0){

25         x = 1;

26         y = 0;

27         return a;

28     }

29     else{

30         long long temp = gcd(b, a % b, x, y);

31         long long t = y;

32         y = x - y * (a / b);

33         x = t;

34         return temp;

35     }

36 }

37 int main()

38 {

39 #ifdef OPEN_FILE

40     freopen("in.txt", "r", stdin);

41     freopen("out.txt", "w", stdout);

42 #endif // OPEN_FILE

43     int T, n;

44     scanf("%d", &T);

45     f[1] = 1;

46     for(long long i = 2; i <= MAXN; i++){

47         f[i] = (f[i - 1] * (4 * i - 2)) % MOD;

48         gcd(i + 1, MOD, x, y);

49         if(x < 0){

50             x = MOD - (-x) % MOD;

51         }

52         x = x % MOD;

53         f[i] = (f[i] * x) % MOD;

54     }

55     for(int cas = 1; cas <= T; cas++){

56         scanf("%d", &n);

57         printf("Case #%d:\n%I64d\n", cas, f[n]);

58     }

59 }

View Code

 中国剩余定理

hdu1573,不互素的中国剩余定理

 1 #include <iostream>

 2 #include <cstdio>

 3 #include <fstream>

 4 #include <algorithm>

 5 #include <cmath>

 6 #include <deque>

 7 #include <vector>

 8 #include <queue>

 9 #include <string>

10 #include <cstring>

11 #include <map>

12 #include <stack>

13 #include <set>

14 //#define OPEN_FILE

15 #define INF 0x3f3f3f3f

16 #define M 11

17 using namespace std;

18 

19 long long a[M], b[M];

20 long long x, y;

21 bool flag;

22 long long gcd(long long  a, long long  b)

23 {

24     if (b == 0){

25         return a;

26     }

27     return gcd(b, a % b);

28 }

29 

30 long long exgcd(long long  a, long long  b, long long & x, long long & y)

31 {

32     if (b == 0){

33         x = 1;

34         y = 0;

35         return a;

36     }

37     else{

38         long long  temp = exgcd(b, a % b, x, y);

39         long long  t = y;

40         y = x - y * (a / b);

41         x = t;

42         return temp;

43     }

44 }

45 int main()

46 {

47 #ifdef OPEN_FILE

48     freopen("in.txt", "r", stdin);

49     //freopen("out.txt", "w", stdout);

50 #endif // OPEN_FILE

51     long long  T, d, n, m, t, A, B;

52     scanf("%I64d", &T);

53     for (int cas = 1; cas <= T; cas++){

54         flag = false;

55         scanf("%I64d%I64d", &n, &m);

56         for (int i = 1; i <= m; i++){

57             scanf("%d", &a[i]);

58             //t = (t * a[i]) / gcd(t, a[i]);

59         }

60         for (int i = 1; i <= m; i++){

61             scanf("%I64d", &b[i]);

62         }

63         A = a[1], B = b[1];

64         for (int i = 2; i <= m; i++){

65             d = exgcd(A, a[i], x, y);

66             if ((b[i] - B) % d != 0){

67                 flag = true;

68                 break;

69             }

70             x = (b[i] - B) / d * x;

71             y = a[i] / d;

72             x = (x % y + y) % y;

73             B = x * A + B;

74             A = (A * a[i]) / d;

75             B = (B % A + A) % A;

76         }

77         if (B > n || flag){

78             printf("0\n");

79         }

80         else{

81             t = 1 + (n - B) / A;

82             if (B == 0){

83                 --t;

84             }

85             printf("%I64d\n", t);

86         }

87     }

88 }

View Code

 

欧拉函数

求1-n与n互质的个数

 1 #include <iostream>

 2 #include <cstdio>

 3 #include <fstream>

 4 #include <algorithm>

 5 #include <cmath>

 6 #include <deque>

 7 #include <vector>

 8 #include <queue>

 9 #include <string>

10 #include <cstring>

11 #include <map>

12 #include <stack>

13 #include <set>

14 //#define OPEN_FILE

15 #define INF 0x3f3f3f3f

16 #define MAXN 32770

17 using namespace std;

18 int x, n;

19 

20 int euler(int x){

21      int ans = x, m = n;

22      for(int i = 2; i * i <= x; i++){

23          if(x % i == 0){

24              ans = ans / i * (i - 1);

25              while(x % i == 0){

26                     x /= i;

27              }

28          }

29      }

30      if(x > 1){

31             ans = ans / x * (x - 1);

32      }

33      return ans;

34 }

35 int main()

36 {

37 #ifdef OPEN_FILE

38     freopen("in.txt", "r", stdin);

39     freopen("out.txt", "w", stdout);

40 #endif // OPEN_FILE

41     int t, cnt;

42     scanf("%d", &n);

43     for(int i = 1; i <= n; i++){

44         scanf("%d", &x);

45         printf("%d\n", euler(x));

46     }

47 }

View Code

 

指数循环节