Number theory concept

Author: Mohit Jain

FastExponentiation

@@ -0,0 +1,46 @@
+#include <bits/stdc++.h>
+
+using namespace std;
+
+
+int power_optimised(int a,int n) {
+	int ans = 1;
+
+	while (n > 0) {
+
+     int lastBit = n&1;
+
+     if(lastBit) {
+     	ans = ans*a;
+
+     }
+     a = a*a;
+
+     n = n>>1; 
+
+	}
+
+	return ans;
+
+}
+
+int main() {
+
+   #ifndef ONLINE_JUDGE
+    // for getting input from input.txt
+    freopen("input.txt", "r", stdin);
+    // for writing output to output.txt
+    freopen("output.txt", "w", stdout);
+  #endif
+    
+    int a,n;
+
+    cin>>a>>n;
+
+    cout<<power_optimised(a,n)<<endl;
+
+
+
+
+	return 0;
+}

FastExponentiation.cpp

@@ -0,0 +1,62 @@
+#include <bits/stdc++.h>
+
+using namespace std;
+
+
+void printFactors(int n) {
+   
+    vector<pair<int, int> > factors;
+  int cnt;
+
+	for(int i =2;i*i <=n; i++) {
+    
+
+       if(n % i == 0) {
+          cnt = 0;
+
+          while( n % i == 0) {
+          	cnt++;
+
+          	n = n/i;
+          }
+           factors.push_back(make_pair(i,cnt));
+       }
+	}
+	if(n!= 1) {
+		factors.push_back(make_pair(n,1));
+
+	}
+
+	for(auto p : factors) {
+
+		cout<< p.first << "^" << p.second << endl;
+	}
+
+	return;
+}
+int main() {
+
+  
+   #ifndef ONLINE_JUDGE
+    // for getting input from input.txt
+    freopen("input.txt", "r", stdin);
+    // for writing output to output.txt
+    freopen("output.txt", "w", stdout);
+  #endif
+
+    int n;
+
+    cin>>n;
+
+    printFactors(n);
+
+
+
+
+
+
+
+
+
+	return 0;
+}

PrimeFactorizationRoot

@@ -0,0 +1,66 @@
+// Sieve Approach - generate an array 
+//containing prime numbers
+
+#include <bits/stdc++.h>
+
+using namespace std;
+
+void prime_sieve(int *p) {
+
+ // first mark all odd numbers as prime
+   for(int i = 3; i < 1000000; i+=2 ) {
+   	p[i] = 1;
+   }
+
+   for(long long i =3; i <= 1000000; i +=2) {
+
+    if(p[i] == 1) {
+
+
+    	for(long long j = i*i; j <= 1000000 ; j += i) {
+           p[j] = 0;
+           
+    	}
+    }
+
+
+
+   }
+
+   // special case
+
+
+   p[1] = p[0] = 0;
+   p[2] = 1;
+
+}
+
+
+
+int main() {
+
+
+   #ifndef ONLINE_JUDGE
+    // for getting input from input.txt
+    freopen("input.txt", "r", stdin);
+    // for writing output to output.txt
+    freopen("output.txt", "w", stdout);
+  #endif
+    
+  int n; 
+
+  cin>>n;
+
+  int p[1000005] = {0};
+
+   prime_sieve(p);
+  for(int i =0 ; i < n ; i ++) {
+
+  	if(p[i] == 1) {
+  		cout<<i<<" ";
+  	}
+  }
+
+
+	return 0;
+}

PrimeFactorizationRoot.cpp

@@ -0,0 +1,50 @@
+// find subsequencof string
+
+
+#include <iostream>
+
+using namespace std;
+
+
+void filterChars(int n, char a[]) {
+
+	int j =0;
+
+	while(n >0) {
+		int lastBit = (n &1);
+		if (lastBit) {
+			cout<<a[j];
+		}
+
+		j++;
+		n = n>>1;
+	}
+
+	cout<<endl;
+}
+void printSubsets(char a[]) {
+
+
+	int n = strlen(a);
+
+	for(int i =1; i < (1<<n) ; i ++) {
+		filterChars(i,a);
+	}
+}
+
+int main() {
+   #ifndef ONLINE_JUDGE
+    // for getting input from input.txt
+    freopen("input.txt", "r", stdin);
+    // for writing output to output.txt
+    freopen("output.txt", "w", stdout);
+  #endif
+  
+	char a[100];
+
+	cin>>a;
+
+	printSubsets(a);
+
+	return 0;
+}

PrimeSieve

@@ -1,2 +1,2 @@
-7
-1 1 1 2 2 2 3
+1
+132

PrimeSieve.cpp

@@ -1 +1,3 @@
-3
+2 
+3 
+11 

SubSequencesBit

@@ -0,0 +1,104 @@
+// useful when we have no of queries to solve
+
+#include <bits/stdc++.h>
+
+using namespace std;
+vector<int> prime_sieve(int *p) {
+
+ // first mark all odd numbers as prime
+   for(int i = 3; i < 1000000; i+=2 ) {
+   	p[i] = 1;
+   }
+
+   for(long long i =3; i <= 1000000; i +=2) {
+
+    if(p[i] == 1) {
+
+
+    	for(long long j = i*i; j <= 1000000 ; j += i) {
+           p[j] = 0;
+           
+    	}
+    }
+
+
+
+   }
+
+   // special case
+
+
+   p[1] = p[0] = 0;
+   p[2] = 1;
+    vector<int> primes;
+
+    primes.push_back(2);
+
+    for(int i =3; i <= 1000000; i +=2) {
+    	if(p[i] == 1) {
+    		primes.push_back(i);
+    	}
+    }
+    return primes;
+}
+
+ vector<int> factiorize(int m, vector<int> &primes) {
+
+ 	vector<int> factors;
+ 	factors.clear();
+
+ 	int i =0;
+ 	int p = primes[0];
+
+ 	while(p*p <= m) {
+ 		if(m%p == 0) {
+ 			factors.push_back(p);
+ 			while(m%p == 0) {
+ 				m = m/p;
+ 			}
+ 		}
+ 		i++;
+ 		p = primes[i];
+ 	}
+ 	if(m != 1) {
+ 		factors.push_back(m);
+ 	}
+     return factors;
+ }
+
+int main() {
+   
+
+   #ifndef ONLINE_JUDGE
+    // for getting input from input.txt
+    freopen("input.txt", "r", stdin);
+    // for writing output to output.txt
+    freopen("output.txt", "w", stdout);
+  #endif
+
+   int p[1000005] = {0};
+
+   vector<int> primes = prime_sieve(p);
+
+   int t;
+
+   cin >> t;
+
+   while(t--) {
+
+   	int no;
+
+   	cin>>no;
+
+   	vector<int> factors = factiorize(no, primes);
+
+   	for(auto f : factors) {
+   		cout<< f << " " << endl;
+   	}
+   }
+
+
+
+
+    return 0;	
+}