Modulo in Competitive programming

Hey Everyone 👋👋

In this article, we will understand some really basic modulo operations and applications for the same.

Now, Modulo in mathematics is a simple operation that returns the remainder when one integer is divided by another, simple isn't it, Smarty-pants!

Essentially we can sum it up like this probably :

Dividend = Divisor × Quotient + Remainder 
Example : 
--> 15 = 2 x 7 + 1 
--> 15 mod 2 = 1
Here 1 is the remainder of 15 when divided by 2

Also, the modulo operator in most of the programming languages is denoted by %.

15 % 2 = 1

➕Addition

Addition with modulo is quite simple :

// Addition

(a + b) % mod = ((a % mod) + (b % mod)) % mod

Eg : 

(15 + 3) % 10 = 18 % 10 = 8

Expanding the brackets :

(15 + 3) % 10 
= ((15 % 10) + (3 %10)) % 10 
= (5 + 3) % 10 
= 8 % 10 = 8

❌ Multiplication

Multiplication with modulo is the same as addition

// Multiplication

(a x b) % mod = ((a % mod ) x (b % mod)) % mod

Eg : 

(4 x 12) % 10 
= (48) % 10 
= 8

Expanding the brackets 

(4 x 12) % 10 
= ((4 % 10) x (12 % 10)) % 10
= (4 x 2) % 10
= 8 % 10
= 8

➖ Subtraction

Subtraction with modulo is a bit tricky You can't do subtraction with the mod operator as you do it with addition and multiplication. Subtraction is a bit strange.

// Subtraction

(a - b) % mod = ((a  % mod )  - (b % mod) + mod) % mod

Eg :

(11 - 3) % 10  = 8

Expanding the brackets : 

= ((11 % 10) - (3 % 10) + 10) % 10 
= (1 -3 + 10 ) %10
=  8 % 10 = 8

Now if try to program it, we don't have to add mod every single time. We can do this :

// We can do this 
int ans = ((x - y) % mod);
if(ans< 0) // if x < y then add mod 
  ans+=mod

// or 

int ans = ((x - y) % mod + mod)%mod;  // Slower than above

➗Division

The division under mod operator is not the same as addition and is very different.

(a / b) % mod may NOT be same as ((a % mod)/(b % mod)) % mod

Now, Modular division does not always exist. We can only do modular division if the is modular inverse of the divisor exists.

Modular Multiplicative inverse of two numbers exists only and only if gcd of those two numbers is equal to 1 that they are relatively prime.

Two integers are relatively prime when there are no common factors other than 1.

So to sum this up :

a / b  = a x 1/b 

(a / b) % mod = (a x (inverse of b if exists)) % mod

Here is the program for the same :


#include<bits/stdc++.h>
using namespace std; 

#define mod 23

int bin_expo(int a, int b){
    int ans = 1;
    while (b>0){
        if (b & 1) // if b is odd 
            ans = (ans *a) % mod;
        a = (a *a) % mod ;
        b = b / 2;
    }
    return ans;
}

int modInverse(int a){ // inverse by fermats theorem
    int g = __gcd(a, mod);
    if (g != 1)
        return -1;
    else{
        // If a and m are relatively prime, then modulo
        // inverse is a^(m-2) mode m
        return bin_expo(a,mod-2);
    }
}

int modDivide(int a, int b){
    a = a % mod;
    int inv = modInverse(b); 
    if (inv == -1)
       return (-1);
    return (inv * a) % mod;
}

signed main(){
    int  a  = 8, b = 3;
    int ans = modDivide(a, b);
    return cout<<ans<<endl,0;    
}

🎯 Final Thoughts

Modulus is crucial and a necessary operator in competitive programming. If you know this it can help you with multiple questions.

Here is a good question if you want to get a hang of it.

Also, do watch this explaination from Errichto, this is where I learned it from.

Congrats🎉🎉 on making it to the end of the article! Have a good day! Stay tuned for such content! 👋✌️

Modulo in Competitive Programming - 1e9 + 7

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