You are given an positive integer n. You have to find the value of (1n +2n + 3n + 4n ) mod 5.
Note : Value of n may be very large of order 1015.
Examples:
Input : n = 4
Output : 4
Explanation : (14 + 24 + 34 + 44)mod 5 = (1+16+81+256)mod 5 = 354 mod 5 = 4Input : n = 2
Output : 0
Explanation : (12 + 22 + 32 + 42)mod 5 = (1+4+9+16)mod 5 = 30 mod 5 = 0
Basic Approach : If you will solve this question with a very basic approach of finding value of (1n +2n + 3n + 4n ) and then finding its modulo value for 5, you will certainly get your answer but for the larger value of n we must got wrong answer as you will be unable to store value of (1n +2n + 3n + 4n ) properly.
Better and Proper Approach : Before proceeding to solution lets go through some of periodical properties of power of 2, 3 & 4.
- f(n) = 2n is periodical for n = 4 in terms of last digit. i.e. last digit of 2n always repeat for next 4th value of n. (ex: 2, 4, 8, 16, 32, 64…)
- f(n) = 3n is periodical for n = 4 in terms of last digit. i.e. last digit of 3n always repeat for next 4th value of n.(ex: 3, 9, 27, 81, 243, 729…)
- f(n) = 4n is periodical for n = 2 in terms of last digit. i.e. last digit of 4n always repeat for next 2nd value of n.(ex: 4, 16, 64, 256..)
- 1n is going to be 1 always, independent of n.
So, If we will have a close look for periodicity of f(n) = (1n +2n + 3n + 4n ) we will get that its periodicity is also 4 and its last digits occurs as :
- for n = 1, f(n) = 10
- for n = 2, f(n) = 30
- for n = 3, f(n) = 100
- for n = 4, f(n) = 354
- for n = 5, f(n) = 1300
Observing above periodicity we can see that if (n%4==0) result of f(n)%5 is going to be 4 other wise result = 0. So, rather than calculating actual value of f(n) and then obtaining its value with mod 5 we can easily get result only by examine value of n.
// Program to find value of f(n)%5 #include <bits/stdc++.h> using namespace std;
// function for obtaining remainder int fnMod( int n)
{ // calculate res based on value of n
return (n % 4) ? 0 : 4;
} // driver program int main()
{ int n = 43;
cout << fnMod(n) << endl;
n = 44;
cout << fnMod(n);
return 0;
} |
// Program to find value of f(n)% 5 class GFG
{ // function for obtaining remainder
static int fnMod( int n)
{
// calculate res based on value of n
return (n % 4 != 0 ) ? 0 : 4 ;
}
// Driver code
public static void main (String[] args)
{
int n = 43 ;
System.out.println(fnMod(n));
n = 44 ;
System.out.print(fnMod(n));
}
} // This code is contributed by Anant Agarwal. |
# program to find f(n) mod 5 def fnMod (n):
res = 4 if (n % 4 = = 0 ) else 0
return res
# driver section n = 43
print (fnMod(n))
n = 44
print (fnMod(n))
|
// C# Program to find value of f(n) % 5 using System;
class GFG {
// function for obtaining remainder
static int fnMod( int n)
{
// calculate res based on value of n
return (n % 4 != 0) ? 0 : 4;
}
// Driver code
public static void Main ()
{
int n = 43;
Console.WriteLine(fnMod(n));
n = 44;
Console.Write(fnMod(n));
}
} // This code is contributed by nitin mittal. |
<?php // PHP Program to find value of f(n)%5 // function for obtaining remainder function fnMod( $n )
{ // calculate res based
// on value of n
return ( $n % 4) ? 0 : 4;
} // Driver Code { $n = 43;
echo fnMod( $n ), "\n" ;
$n = 44;
echo fnMod( $n );
return 0;
} // This code is contributed by nitin mittal. ?> |
<script> // JavaScript program to find value of f(n)% 5 // function for obtaining remainder function fnMod(n)
{
// calculate res based on value of n
return (n % 4 != 0) ? 0 : 4;
}
// Driver Code let n = 43;
document.write(fnMod(n) + "<br/>" );
n = 44;
document.write(fnMod(n) + "<br/>" );
// This code is contributed by splevel62. </script> |
0 4
Time complexity: O(1)
Auxiliary space: O(1)