数论比赛

Problem A Play with Floor and Ceil http://uva.onlinejudge.org/index.php?option=com_onlinejudge&Itemid=8&category=18&page=show_problem&problem=1614

扩展gcd求解2元不定式 ax+by=c    d=gcd(a,b)=ax+by;   x=x*c/d  y=y*c/d

#include<cstdio>
typedef long long LL;
LL ext_gcd(LL a,LL b,LL &x,LL &y) { //扩展gcd d=gcd(a,b)=a*x+b*y; return d,x,y;
    LL t,ret;
    if(!b) {
        x=1;
        y=0;
        return a;
    }
    ret=ext_gcd(b,a%b,x,y);
    t=x;
    x=y;
    y=t-a/b*y;
    return ret;
}
int main(){
    int t,x,k;
    while(~scanf("%d",&t)){
        while(t--){
            scanf("%d%d",&x,&k);
            LL a=x/k;
            LL b=a;
            if(x%k) b++;
            LL tx,ty;
            LL d=ext_gcd(a,b,tx,ty);
            printf("%lld %lld\n",tx*x/d,ty*x/d);
        }
    }
    return 0;
}

 Alternate Task http://uva.onlinejudge.org/index.php?option=com_onlinejudge&Itemid=8&category=117&page=show_problem&problem=2828

求因子的和

#include<cstdio>
int ds_func(int n){//求n所有除数的和
    int ret=1,m=n,t;
    for(int i=2;i*i<=n;i+=(i==2)?1:2){
        if(!(n%i)){
            t=i*i;
            n/=i;
            while(!(n%i)){
                t*=i;
                n/=i;
            }
            ret*=(t-1)/(i-1);
        }
    }
    return n>1?ret*(n+1):ret;
}
int main(){
    int s,cas=1;
    while(~scanf("%d",&s),s){
        int ans=-1;
        for(int i=s;i>=1;i--){
            if(ds_func(i)==s){
                ans=i;
                break;
            }
        }
        printf("Case %d: %d\n",cas++,ans);
    }
    return 0;
}

 

1434 - YAPTCHA  http://uva.onlinejudge.org/index.php?option=com_onlinejudge&Itemid=8&page=show_problem&category=447&problem=4180&mosmsg=Submission+received+with+ID+14036675

找规律，发现3*k+7是素数值为1，否则为0，素数筛法。

#include<cstdio>
#include<cstring>
#define mt(a,b) memset(a,b,sizeof(a))
const int M=3000010;
int pri[M],mark[M],pricnt;//mark[i]存i的最小因子，素数时mark[i]==i
void sieve_primes() { //筛素数
    pricnt=0;
    mt(mark,0);
    mark[0]=mark[1]=1;
    for(int i=2; i<M; i++) {
        if(!mark[i]) pri[pricnt++]=mark[i]=i;
        for(int j=0; pri[j]*i<M; j++) {
            mark[i*pri[j]]=pri[j];
            if(!(i%pri[j])) break;
        }
    }
}
int ans[M];
int main(){
    sieve_primes();
    ans[0]=0;
    for(int i=1;i<=1000000;i++){
        int j=3*i+7;
        ans[i]=ans[i-1];
        if(mark[j]==j) ans[i]++;
    }
    int t,n;
    while(~scanf("%d",&t)){
        while(t--){
            scanf("%d",&n);
            printf("%d\n",ans[n]);
        }
    }
    return 0;
}

 

 

1415 - Gauss Prime http://uva.onlinejudge.org/index.php?option=com_onlinejudge&Itemid=8&category=447&page=show_problem&problem=4161

#include<cstdio>
typedef long long LL;
LL mulmod(LL a,LL b,LL c) { //ret=(a*b)%c
    LL ret=0;
    for(; b; a=(a<<1)%c,b>>=1) {
        if(b&1) {
            ret=(ret+a)%c;
        }
    }
    return ret;
}
LL powmod(LL a,LL b,LL c) { //ret=(a^b)%mod
    LL ret=1%c;
    for(; b; a=mulmod(a,a,c),b>>=1) {
        if(b&1) {
            ret=mulmod(ret,a,c);
        }
    }
    return ret;
}
bool suspect(LL a,int s,LL d,LL n) {
    LL x=powmod(a,d,n);
    if(x==1) return true;
    while(s--) {
        if(x==n-1) return true;
        x=mulmod(x,x,n);
    }
    return false;
}
const int test[]= {2,3,5,7,11,13,17,19,23,29,-1}; // for n < 10^16
bool isprime(LL n) { //Miller-Rabin 大素数测试
    if(n<=1||(n>2&&(!(n&1)))) return false;
    LL d=n-1;
    int s=0;
    while(!(d&1)) {
        s++;
        d>>=1;
    }
    for(int i=0; test[i]<n&&~test[i]; i++) {
        if(!suspect(test[i],s,d,n)) return false;
    }
    return true;
}
int main(){
    int t,a,b;
    while(~scanf("%d",&t)){
        while(t--){
            scanf("%d%d",&a,&b);
            if(a){
                LL t=a*a+2*b*b;
                if(isprime(t)){
                    puts("Yes");
                    continue;
                }
            }
            puts("No");
        }
    }
    return 0;
}

 

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