Category Modular Arithmatic

Bigmod & mod_inverse

Arunima Mandal Leave a comment 
#include <bits/stdc++.h>

#define pf                  printf
#define sf(a)               scanf("%d",&a)
#define sfl(a)              scanf("%lld",&a)
#define sff(a,b)            scanf("%d %d",&a,&b)
#define sffl(a,b)           scanf("%lld %lld",&a,&b)
#define sfff(a,b,c)         scanf("%d %d %d",&a,&b,&c)
#define sfffl(a,b,c)        scanf("%lld %lld %lld",&a,&b,&c)
#define sffff(a,b,c,d)      scanf("%d %d %d %d",&a,&b,&c,&d)
#define sffffl(a,b,c,d)     scanf("%lld %lld %lld %lld",&a,&b,&c,&d)
#define sfffff(a,b,c,d,e)   scanf("%d %d %d %d %d",&a,&b,&c,&d,&e)
#define sfffffl(a,b,c,d,e)  scanf("%lld %lld %lld %lld %lld",&a,&b,&c,&d,&e)
#define ms(a,b)             memset(a,b,sizeof(a))
#define pb(a)               push_back(a)
#define db                  double
#define ft                  float
#define ll                  long long
#define ull                 unsigned long long
#define ff                  first
#define ss                  second
#define sz(x)               x.size()
#define qu                  queue
#define pqu                 priority_queue
#define vc                  vector
#define vi                  vector<int>
#define vll                 vector<long long>
#define pii                 pair<int,int>
#define pis                 pair<int,string>
#define psi                 pair<string,int>
#define all(x)              x.begin(),x.end()
#define CIN                 ios_base::sync_with_stdio(0); cin.tie(0)
#define max3(a, b, c)       max(a, b) > max(b, c) ? max(a, b) : max(b, c)
#define min3(a, b, c)       min(a, b) < min(b, c) ? min(a, b) : min(b, c)
#define loop0(i,n)          for(int i=0;i<n;i++)
#define loop1(i,n)          for(int i=1;i<=n;i++)
#define loopab(a,b)         for(int i=a;i<=b;i++)
#define stlloop(x)          for(__typeof(x.begin()) it=x.begin();it!=x.end();it++)
#define gcd(a, b)           __gcd(a, b)
#define lcm(a, b)           ((a)*((b)/gcd(a,b)))
#define case1(z)            cout<<"Case "<<z<<": "
#define case2(z)            printf("Case %d: ",z)
#define PI                  3.14159265358979323846264338328
#define valid(tx,ty)        tx>=0 && tx<r && ty>=0 && ty<c
#define intlim              2147483648
#define MAX                 1000000
#define inf                 10000000

/*------------------------------Graph Moves----------------------------*/
//const int fx[]={+1,-1,+0,+0};
//const int fy[]={+0,+0,+1,-1};
//const int fx[]={+0,+0,+1,-1,-1,+1,-1,+1};   // Kings Move
//const int fy[]={-1,+1,+0,+0,+1,+1,-1,-1};  // Kings Move
//const int fx[]={-2, -2, -1, -1,  1,  1,  2,  2};  // Knights Move
//const int fy[]={-1,  1, -2,  2, -2,  2, -1,  1}; // Knights Move
/*---------------------------------------------------------------------*/

using namespace std;

ll m;

ll bigmod(ll n,ll p)
{
    if(p==0) return 1;
    if(p%2==0)
    {
        ll x=bigmod(n,p/2)%m;
        return (x*x)%m;
    }
    else return ((n%m)*(bigmod(n,p-1)%m))%m;
}


ll modinverse(ll n)
{
    return bigmod(n,m-2);
}

int main()
{
    //CIN;
    //freopen("in.txt","r",stdin);
    //freopen("out.txt","w",stdout);
    ll a,b;
    while(cin>>a>>b>>m)
    {
        cout<<bigmod(a,b)<<endl;
    }
    return 0;
}

UVa – 10127 – Ones

Arunima Mandal Leave a comment 
#include <bits/stdc++.h>

#define sf              scanf
#define pf              printf
#define ms(a,b)         memset(a,b,sizeof(a))
#define pb(a)           push_back(a)
#define db              double
#define ft              float
#define ll              long long
#define ull             unsigned long long
#define ff              first
#define ss              second
#define sz(x)           x.size()
#define qu              queue
#define pqu             priority_queue
#define vc              vector
#define vi              vector<int>
#define vll             vector<long long>
#define pii             pair<int,int>
#define pis             pair<int,string>
#define psi             pair<string,int>
#define all(x)          x.begin(),x.end()
#define CIN             ios_base::sync_with_stdio(0); cin.tie(0)
#define loop0(i,n)      for(int i=0;i<n;i++)
#define loop1(i,n)      for(int i=1;i<=n;i++)
#define stlloop(x)     for(__typeof(x.begin()) it=x.begin();it!=x.end();it++)
#define gcd(a, b)       __gcd(a, b)
#define lcm(a, b)       ((a)*((b)/gcd(a,b)))
#define case(z,x)       cout<<"Case "<<i<<": "<<x<<endl
#define case(z)         cout<<"Case "<<z<<": "
#define PI              3.14159265358979323846264338328
#define valid(tx,ty)    tx>=0 && tx<r && ty>=0 && ty<c
#define MAX             2000
#define inf             10000000

/*------------------------------Graph Moves----------------------------*/
//const int fx[]={+1,-1,+0,+0};
//const int fy[]={+0,+0,+1,-1};
//const int fx[]={+0,+0,+1,-1,-1,+1,-1,+1};   // Kings Move
//const int fy[]={-1,+1,+0,+0,+1,+1,-1,-1};  // Kings Move
//const int fx[]={-2, -2, -1, -1,  1,  1,  2,  2};  // Knights Move
//const int fy[]={-1,  1, -2,  2, -2,  2, -1,  1}; // Knights Move
/*---------------------------------------------------------------------*/

using namespace std;

int m;
int bigmod(int a,int p)
{
    if(p==0) return 1;
    if(p%2==0)
    {
        return ((bigmod(a,p/2)%m)*(bigmod(a,p/2)%m))%m;
    }
    else
        return ((a%m)*(bigmod(a,p-1)%m))%m;
}

int main()
{
    CIN;
//    freopen("in.txt","r",stdin);
//    freopen("out.txt","w",stdout);

    while(cin>>m)
    {
        int x=1,c=1;
        while(x!=0)
        {
            x=(x*10)+1;
            c++;
            x=bigmod(x,1);
        }
        cout<<c<<endl;
    }
    return 0;
}

UVa – 374 – Bigmod

Arunima Mandal Leave a comment 
//#include <bits/stdc++.h>
//
//#define ms(a,b)         memset(a,b,sizeof(a))
//#define pb(a)           push_back(a)
//#define db              double
//#define ft              float
//#define ll              long long
//#define ull             unsigned long long
//#define ff              first
//#define ss              second
//#define sz(x)           x.size()
//#define qu              queue
//#define pqu             priority_queue
//#define vc              vector
//#define vi              vector<int>
//#define vll             vector<long long>
//#define pii             pair<int,int>
//#define pis             pair<int,string>
//#define psi             pair<string,int>
//#define all(x)          x.begin(),x.end()
//#define CIN             ios_base::sync_with_stdio(0); cin.tie(0)
//#define loop0(i,n)      for(int i=0;i<n;i++)
//#define loop1(i,n)      for(int i=1;i<=n;i++)
//#define stlloop(x)     for(__typeof(x.begin()) it=x.begin();it!=x.end();it++)
//#define gcd(a, b)       __gcd(a, b)
//#define lcm(a, b)       ((a)*((b)/gcd(a,b)))
//#define case(z,x)       cout<<"Case "<<i<<": "<<x<<endl
//#define case(z)         cout<<"Case "<<z<<": "
//#define PI              3.14159265358979323846264338328
//#define valid(tx,ty)    tx>=0 && tx<r && ty>=0 && ty<c
//#define MAX             2000
//#define inf             10000000
//
///*----------------------Graph Moves----------------*/
////const int fx[]={+1,-1,+0,+0};
////const int fy[]={+0,+0,+1,-1};
////const int fx[]={+0,+0,+1,-1,-1,+1,-1,+1};   // Kings Move
////const int fy[]={-1,+1,+0,+0,+1,+1,-1,-1};  // Kings Move
////const int fx[]={-2, -2, -1, -1,  1,  1,  2,  2};  // Knights Move
////const int fy[]={-1,  1, -2,  2, -2,  2, -1,  1}; // Knights Move
///*------------------------------------------------*/
//
//using namespace std;
//
//ll big_mod(ll n, ll pow, ll mod)
//{
//    if(pow==0) return 1;
//    if(pow==1) return n;
//
//    ll ret=0;
//
//    if(pow%2==0)
//    {
//        ret=big_mod(n,pow/2,mod)%mod;
//        return (ret*ret)%mod;
//    }
//    else
//        return ((n%mod)*(big_mod(n,pow-1,mod))%mod)%mod;
//
//}
//
//int main()
//{
//    CIN;
//    //freopen("out.txt","w",stdout);
//    ll a,b,c;
//    cin>>a>>b>>c;
//    cout<<big_mod(a,b,c)<<endl;
////    ll d=1;
////    for(int i=0;i<b;i++)
////    {
////        d*=a;
////        d%=c;
////    }
////    cout<<d<<endl;
//
//    return 0;
//}




#include <bits/stdc++.h>

#define ms(a,b)         memset(a,b,sizeof(a))
#define pb(a)           push_back(a)
#define db              double
#define ft              float
#define ll              long long
#define ull             unsigned long long
#define ff              first
#define ss              second
#define sz(x)           x.size()
#define qu              queue
#define pqu             priority_queue
#define vc              vector
#define vi              vector<int>
#define vll             vector<long long>
#define pii             pair<int,int>
#define pis             pair<int,string>
#define psi             pair<string,int>
#define all(x)          x.begin(),x.end()
#define CIN             ios_base::sync_with_stdio(0); cin.tie(0)
#define loop0(i,n)      for(int i=0;i<n;i++)
#define loop1(i,n)      for(int i=1;i<=n;i++)
#define stlloop(x)     for(__typeof(x.begin()) it=x.begin();it!=x.end();it++)
#define gcd(a, b)       __gcd(a, b)
#define lcm(a, b)       ((a)*((b)/gcd(a,b)))
#define case(z,x)       cout<<"Case "<<i<<": "<<x<<endl
#define case(z)         cout<<"Case "<<z<<": "
#define PI              3.14159265358979323846264338328
#define valid(tx,ty)    tx>=0 && tx<r && ty>=0 && ty<c
#define MAX             2000
#define inf             10000000

/*----------------------Graph Moves----------------*/
//const int fx[]={+1,-1,+0,+0};
//const int fy[]={+0,+0,+1,-1};
//const int fx[]={+0,+0,+1,-1,-1,+1,-1,+1};   // Kings Move
//const int fy[]={-1,+1,+0,+0,+1,+1,-1,-1};  // Kings Move
//const int fx[]={-2, -2, -1, -1,  1,  1,  2,  2};  // Knights Move
//const int fy[]={-1,  1, -2,  2, -2,  2, -1,  1}; // Knights Move
/*------------------------------------------------*/

using namespace std;

ll a,b,m;
ll bigmod(ll n,ll p)
{
    if(p==0) return 1;
    if(p%2==0)
    {
        return ((bigmod(n,p/2)%m)*(bigmod(n,p/2)%m))%m;
    }
    else
        return ((n%m)*(bigmod(n,p-1)%m))%m;
}

int main()
{
    CIN;
    freopen("in.txt","r",stdin);
    freopen("out.txt","w",stdout);

    while(cin>>a>>b>>m)
    {
        cout<<bigmod(a,b)<<endl;
    }
    return 0;
}

Light OJ – 1067 – Combinations

Arunima Mandal Leave a comment 
#include &lt;bits/stdc++.h&gt;

#define pf                  printf
#define sf(a)               scanf(&quot;%d&quot;,&amp;a)
#define sfl(a)              scanf(&quot;%lld&quot;,&amp;a)
#define sff(a,b)            scanf(&quot;%d %d&quot;,&amp;a,&amp;b)
#define sffl(a,b)           scanf(&quot;%lld %lld&quot;,&amp;a,&amp;b)
#define sfff(a,b,c)         scanf(&quot;%d %d %d&quot;,&amp;a,&amp;b,&amp;c)
#define sfffl(a,b,c)        scanf(&quot;%lld %lld %lld&quot;,&amp;a,&amp;b,&amp;c)
#define sffff(a,b,c,d)      scanf(&quot;%d %d %d %d&quot;,&amp;a,&amp;b,&amp;c,&amp;d)
#define sffffl(a,b,c,d)     scanf(&quot;%lld %lld %lld %lld&quot;,&amp;a,&amp;b,&amp;c,&amp;d)
#define sfffff(a,b,c,d,e)   scanf(&quot;%d %d %d %d %d&quot;,&amp;a,&amp;b,&amp;c,&amp;d,&amp;e)
#define sfffffl(a,b,c,d,e)  scanf(&quot;%lld %lld %lld %lld %lld&quot;,&amp;a,&amp;b,&amp;c,&amp;d,&amp;e)
#define ms(a,b)             memset(a,b,sizeof(a))
#define pb(a)               push_back(a)
#define db                  double
#define ft                  float
#define ll                  long long
#define ull                 unsigned long long
#define ff                  first
#define ss                  second
#define sz(x)               x.size()
#define qu                  queue
#define pqu                 priority_queue
#define vc                  vector
#define vi                  vector&lt;int&gt;
#define vll                 vector&lt;long long&gt;
#define pii                 pair&lt;int,int&gt;
#define pis                 pair&lt;int,string&gt;
#define psi                 pair&lt;string,int&gt;
#define all(x)              x.begin(),x.end()
#define CIN                 ios_base::sync_with_stdio(0); cin.tie(0)
#define max3(a, b, c)       max(a, b) &gt; max(b, c) ? max(a, b) : max(b, c)
#define min3(a, b, c)       min(a, b) &lt; min(b, c) ? min(a, b) : min(b, c)
#define loop0(i,n)          for(int i=0;i&lt;n;i++)
#define loop1(i,n)          for(int i=1;i&lt;=n;i++)
#define loopab(a,b)         for(int i=a;i&lt;=b;i++)
#define stlloop(x)          for(__typeof(x.begin()) it=x.begin();it!=x.end();it++)
#define gcd(a, b)           __gcd(a, b)
#define lcm(a, b)           ((a)*((b)/gcd(a,b)))
#define case1(z)            cout&lt;&lt;&quot;Case &quot;&lt;&lt;z&lt;&lt;&quot;: &quot;
#define case2(z)            printf(&quot;Case %d: &quot;,z)
#define PI                  3.14159265358979323846264338328
#define valid(tx,ty)        tx&gt;=0 &amp;&amp; tx&lt;r &amp;&amp; ty&gt;=0 &amp;&amp; ty&lt;c
#define intlim              2147483648
#define MAX                 1000000
#define inf                 10000000

/*------------------------------Graph Moves----------------------------*/
//const int fx[]={+1,-1,+0,+0};
//const int fy[]={+0,+0,+1,-1};
//const int fx[]={+0,+0,+1,-1,-1,+1,-1,+1};   // Kings Move
//const int fy[]={-1,+1,+0,+0,+1,+1,-1,-1};  // Kings Move
//const int fx[]={-2, -2, -1, -1,  1,  1,  2,  2};  // Knights Move
//const int fy[]={-1,  1, -2,  2, -2,  2, -1,  1}; // Knights Move
/*---------------------------------------------------------------------*/

using namespace std;

ll m=1000003;

ll bigmod(ll n,ll p)
{
    if(p==0) return 1;
    if(p%2==0)
    {
        ll x=bigmod(n,p/2)%m;
        return (x*x)%m;
    }
    else return ((n%m)*bigmod(n,p-1)%m)%m;
}

ll mod_inverse(ll n)
{
    return bigmod(n,m-2);
}

ll dp[1000006];

void fact()
{
    ll x=1;
    for(int i=1; i&lt;=1000002; i++)
    {
        x=(x%m*i%m)%m;
        dp[i]=x;
    }
}

int main()
{
    //CIN;
    //freopen(&quot;in.txt&quot;,&quot;r&quot;,stdin);
    //freopen(&quot;out.txt&quot;,&quot;w&quot;,stdout);
    int t;
    sf(t);
    fact();
    loop1(z,t)
    {
        ll n,r;
        sffl(n,r);
        case2(z);
        if(n==r || r==0)
        {
            pf(&quot;1\n&quot;);
        }
        else
        {
            ll p=(dp[r]%m*dp[n-r]%m)%m;
            ll a=mod_inverse(p);
            ll b=(dp[n]%m*a%m)%m;
            pf(&quot;%lld\n&quot;,b);
        }
    }
    return 0;
}

Light OJ- 1054 – Efficient Pseudo Code

Arunima Mandal Leave a comment 
#include <bits/stdc++.h>

#define pf                  printf
#define sf(a)               scanf("%d",&a)
#define sfl(a)              scanf("%lld",&a)
#define sff(a,b)            scanf("%d %d",&a,&b)
#define sffl(a,b)           scanf("%lld %lld",&a,&b)
#define sfff(a,b,c)         scanf("%d %d %d",&a,&b,&c)
#define sfffl(a,b,c)        scanf("%lld %lld %lld",&a,&b,&c)
#define sffff(a,b,c,d)      scanf("%d %d %d %d",&a,&b,&c,&d)
#define sffffl(a,b,c,d)     scanf("%lld %lld %lld %lld",&a,&b,&c,&d)
#define sfffff(a,b,c,d,e)   scanf("%d %d %d %d %d",&a,&b,&c,&d,&e)
#define sfffffl(a,b,c,d,e)  scanf("%lld %lld %lld %lld %lld",&a,&b,&c,&d,&e)
#define ms(a,b)             memset(a,b,sizeof(a))
#define pb(a)               push_back(a)
#define db                  double
#define ft                  float
#define ll                  long long
#define ull                 unsigned long long
#define ff                  first
#define ss                  second
#define sz(x)               x.size()
#define qu                  queue
#define pqu                 priority_queue
#define vc                  vector
#define vi                  vector<int>
#define vll                 vector<long long>
#define pii                 pair<int,int>
#define pis                 pair<int,string>
#define psi                 pair<string,int>
#define all(x)              x.begin(),x.end()
#define CIN                 ios_base::sync_with_stdio(0); cin.tie(0)
#define max3(a, b, c)       max(a, b) > max(b, c) ? max(a, b) : max(b, c)
#define min3(a, b, c)       min(a, b) < min(b, c) ? min(a, b) : min(b, c)
#define loop0(i,n)          for(int i=0;i<n;i++)
#define loop1(i,n)          for(int i=1;i<=n;i++)
#define loopab(a,b)         for(int i=a;i<=b;i++)
#define stlloop(x)          for(__typeof(x.begin()) it=x.begin();it!=x.end();it++)
#define gcd(a, b)           __gcd(a, b)
#define lcm(a, b)           ((a)*((b)/gcd(a,b)))
#define case1(z)            cout<<"Case "<<z<<": "
#define case2(z)            printf("Case %d: ",z)
#define PI                  3.14159265358979323846264338328
#define valid(tx,ty)        tx>=0 && tx<r && ty>=0 && ty<c
#define intlim              2147483648
#define MAX                 100000
#define inf                 10000000

/*------------------------------Graph Moves----------------------------*/
//const int fx[]={+1,-1,+0,+0};
//const int fy[]={+0,+0,+1,-1};
//const int fx[]={+0,+0,+1,-1,-1,+1,-1,+1};   // Kings Move
//const int fy[]={-1,+1,+0,+0,+1,+1,-1,-1};  // Kings Move
//const int fx[]={-2, -2, -1, -1,  1,  1,  2,  2};  // Knights Move
//const int fy[]={-1,  1, -2,  2, -2,  2, -1,  1}; // Knights Move
/*---------------------------------------------------------------------*/

using namespace std;

bool p[MAX+2];
vector<int>prime;

void sieve()
{
    p[0]=p[1]=1;
    int root=sqrt(MAX);
    for(int i=2; i<=root; i++)
    {
        if(p[i]==0)
        {
            //prime.push_back(i);
            for(int j=i*i; j<=MAX; j+=i)
            {
                p[j]=1;
                if(i%2==1) j+=i;
            }
        }
    }
    for(int i=2; i<=MAX; i++)
    {
        if(p[i]==0) prime.push_back(i);
    }
}

ll m=1000000007;

ll big_mod(ll n,ll p)
{
    if(p==0) return 1;
    if(p%2==0)
    {
        ll x=big_mod(n,p/2)%m;
        return (x*x)%m;
    }
    else return ((n%m)*(big_mod(n,p-1)%m))%m;
}

ll mod_inverse(ll n)
{
    return big_mod(n,m-2);
}

ll sum_of_divisor(ll n,ll p)
{
    ll sum=1,c,s;
    ll k=n;
    ll root=sqrt(n);
    for(ll i=0; prime[i]<=root; i++)
    {
        if(k%prime[i]==0)
        {
            c=0;
            while(k%prime[i]==0)
            {
                k/=prime[i];
                c++;
            }
            root=sqrt(k);
            ll x=big_mod(prime[i],(c*p)+1);
            ll y=mod_inverse(prime[i]-1);
            s=(((x-1+m)%m)*((y+m)%m))%m; // here we add m to get the exact result because x or y can be of negative value since in the problem, n & m are signed integer
            sum=((sum%m)*(s%m))%m;
        }
    }
    if(k>1)
    {
        ll x=big_mod(k,p+1);
        ll y=mod_inverse(k-1);
        s=(((x-1+m)%m)*((y+m)%m))%m;
        sum=((sum%m)*(s%m))%m;
    }
    return sum;
}

int main()
{
    //CIN;
//    freopen("in.txt","r",stdin);
//    freopen("out.txt","w",stdout);
    int t;
    sf(t);
    sieve();
    loop1(z,t)
    {
        ll n,p;
        sffl(n,p);
        case2(z);
        pf("%lld\n",sum_of_divisor(n,p));
    }
    return 0;
}