From 975b3dc27e0f30b7e476e7eb4cdcf7626e8d796f Mon Sep 17 00:00:00 2001
From: Lavish Saluja <lavishsaluja.ls@gmail.com>
Date: Tue, 12 Jun 2018 16:22:21 +0530
Subject: [PATCH 1/3] Added SPOJ

---
 Problem Solutions/SPOJ/thelastdigit.cpp | 30 +++++++++++++++++++++++++
 1 file changed, 30 insertions(+)
 create mode 100644 Problem Solutions/SPOJ/thelastdigit.cpp

diff --git a/Problem Solutions/SPOJ/thelastdigit.cpp b/Problem Solutions/SPOJ/thelastdigit.cpp
new file mode 100644
index 00000000..111f45f9
--- /dev/null
+++ b/Problem Solutions/SPOJ/thelastdigit.cpp	
@@ -0,0 +1,30 @@
+#include <iostream>
+#include <bits/stdc++.h>
+
+using namespace std;
+
+int main() {
+	int t;
+	scanf("%d",&t);
+	
+	while(t--)
+	{
+		int x,y,res=1;
+		
+		scanf("%d",&x);
+		scanf("%d",&y);
+		
+		x=x%10;
+		
+		while(y>0)
+		{
+			if(y%2!=0)
+			res=(res*x)%10;
+
+			y=y/2;
+			x=(x*x)%10;
+		}
+		
+		printf("%d\n",res);
+	}
+	}
\ No newline at end of file

From 623c389efc9da0ea97b5d00890a36d16af5a50bd Mon Sep 17 00:00:00 2001
From: Lavish Saluja <lavishsaluja.ls@gmail.com>
Date: Tue, 12 Jun 2018 17:22:35 +0530
Subject: [PATCH 2/3] added comments, changed the logic.

---
 Problem Solutions/SPOJ/thelastdigit.cpp | 46 +++++++++++--------------
 1 file changed, 20 insertions(+), 26 deletions(-)

diff --git a/Problem Solutions/SPOJ/thelastdigit.cpp b/Problem Solutions/SPOJ/thelastdigit.cpp
index 111f45f9..ae794a1b 100644
--- a/Problem Solutions/SPOJ/thelastdigit.cpp	
+++ b/Problem Solutions/SPOJ/thelastdigit.cpp	
@@ -1,30 +1,24 @@
 #include <iostream>
-#include <bits/stdc++.h>
-
 using namespace std;
-
+int mod(int,int);//modular function
 int main() {
-	int t;
-	scanf("%d",&t);
-	
-	while(t--)
-	{
-		int x,y,res=1;
-		
-		scanf("%d",&x);
-		scanf("%d",&y);
-		
-		x=x%10;
-		
-		while(y>0)
-		{
-			if(y%2!=0)
-			res=(res*x)%10;
-
-			y=y/2;
-			x=(x*x)%10;
-		}
-		
-		printf("%d\n",res);
+	//will be using modular exponentiation (check wikipedia for details over the algo)
+	int t;cin>>t;
+	while(t--){
+	    int a,b;cin>>a>>b;//we need to find last digit of a^b
+	    cout<<mod(a,b);
+	    cout<<endl;
 	}
-	}
\ No newline at end of file
+	return 0;
+}
+int mod(int a,int b){
+    if(b==0)//power is 0
+        return 1;
+    else if(b%2==0){//power is even
+        int y = mod(a,b/2);
+        return (y*y)%10;
+    }
+    else{//power is odd
+        return ((a%10)*(mod(a,b-1)%10))%10;
+    }
+}

From 50243e732cd3f31a1ee094195e1b79f2ae932c24 Mon Sep 17 00:00:00 2001
From: Lavish Saluja <lavishsaluja.ls@gmail.com>
Date: Wed, 13 Jun 2018 11:39:05 +0530
Subject: [PATCH 3/3] Algo to find all the primes

Adding Sieve of Eratosthenes.
---
 .../FindingAllPrimes/SieveofEratosthenes.cpp  | 33 +++++++++++++++++++
 1 file changed, 33 insertions(+)
 create mode 100644 Algorithms/NumberTheory/FindingAllPrimes/SieveofEratosthenes.cpp

diff --git a/Algorithms/NumberTheory/FindingAllPrimes/SieveofEratosthenes.cpp b/Algorithms/NumberTheory/FindingAllPrimes/SieveofEratosthenes.cpp
new file mode 100644
index 00000000..2afefb3d
--- /dev/null
+++ b/Algorithms/NumberTheory/FindingAllPrimes/SieveofEratosthenes.cpp
@@ -0,0 +1,33 @@
+// Sieve Of Eratosthenes
+// Time Complexity O(nlog(log(n))) whereas following brute force will end you up with O(n*sqrt(n))
+#include <iostream>
+#include<math.h>
+using namespace std;
+
+int main() {
+    int n;cin>>n;
+    int prime[n+1];//prime[i]=1 will imply that i is prime & prime[i]=0 will imply that i is not prime
+    for(int i=0;i<=n;i++)
+        prime[i]=1;//considering all as primes as of now.
+    prime[0]=0;
+    prime[1]=0;//since 0 & 1 are not prime ofcourse
+    
+    /*
+    Here we will be running are first loop only till sqrt(n) instead of n because we know that
+    if a number is prime it does not have any factor till its square root so we
+    can use it for optimization.
+    */
+    for(int i=2;i<=sqrt(n);i++){
+        if(prime[i]==1){//if i is prime
+            for(int j=2;j*i<=n;j++){
+                prime[i*j]=0;//setting the multiples of i to 0 because they can't be prime ofcourse
+            }
+        }
+    }
+    for(int i=0;i<=n;i++){
+        if(prime[i]!=0){
+            cout<<i<<endl;
+        }
+    }
+	return 0;
+}
\ No newline at end of file