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Find value of (n^1 + n^2 + n^3 + n^4) mod 5 for given n in C++
In this problem, we are given a value n. Our task is to find value of (n^1 + n^2 + n^3 + n^4) mod 5 for given n.
Let's take an example to understand the problem,
Input : n= 5 Output : 0
Explanation −
(51 + 52 + 53 + 54) mod 5 = (5 + 25 + 125 + 625) mod 5 = (780) mode 5 = 0
Solution Approach
A simple solution to the problem is by directly finding the value of the equation for the given value of N and then calculating its modulus with 5.
Example
Program to illustrate the working of our solution
#include <iostream>
using namespace std;
int findMod5Val(int n){
int val = (n + (n*n) + (n*n*n) + (n*n*n*n));
return val%5;
}
int main(){
int n = 12;
cout<<"For N = "<<n<<", the value of (n^1 + n^2 + n^3 + n^4)\%5 is "<<findMod5Val(n);
return 0;
}Output
For N = 12, the value of (n^1 + n^2 + n^3 + n^4)%5 is 0
Another solution to the problem is by using mathematical formulation and generalisation of the function.
$\mathrm{f(n)\:=\:(n\:+\:n^2\:+\:n^3\:+\:n^4)}$
$\mathrm{f(n)\:=\:n^*(1\:+\:n\:+\:n^2\:+\:n^3)}$
$\mathrm{f(n)\:=\:n^*(1^*(1+n)+n^{2*}(1+n))}$
$\mathrm{f(n)\:=\:n^*((1+n^2)^*(1+n))}$
$\mathrm{f(n)\:=\:n^*(n+1)^*(n^2+1)}$
For this equation we can derive that the value of f(n) % 5 can be either 0 or 4 based on the value of n.
if(n%5 == 1), f(n)%5 = 4 Else, f(n)%5 = 0
Example
Program to illustrate the working of our solution
#include <iostream>
using namespace std;
int findMod5Val(int n){
if(n % 4 == 1)
return 4;
return 0;
}
int main(){
int n = 65;
cout<<"For N = "<<n<<", the value of (n^1 + n^2 + n^3 + n^4)\%5 is "<<findMod5Val(n);
return 0;
}Output
For N = 65, the value of (n^1 + n^2 + n^3 + n^4)%5 is 4
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