maths.txt

Modular arithmatic 

basically what happens is we have to do certain calculations in our coding and sometimes the result is so large that it cannot be stored even in the 
largest datatypes 

for int max capcity is 10^9 
the largest data type that we know is long long int it can store more powers than int approx 10 ^20

Modulo properties help us in getting the result stored in int or long datatype 
properties :
1. (a+b)%M = ((a+%M)+(b%M))%M  -> the result will come in same as the lhs 
2. (a*b)%m = (()) same but change the sign in btw the two entity 
3. Incase of - you need to add the M and change the sign 
4. (A/b)%M = ((a%M *(b inverse)%M))%m




In THE QUESTION what you need to do is the 
Har step pe % M 
Upar m ko declare karo long long mein then give the value that is given in the qustion  (mostly : 10^9+7 )as this number is very close to Integer Maximum 
 


 GCD (HCF) PROGRAMMING EUCLID 
 overview 
4  and 12 -> 4 
12, 18 -> 6 

how to find gcd ?
prime factorsion karo
select the minimum power among the two digits of every factor 
this is the GCD 


LCM 
max of power baki sab process same 


Intersting properties: a*b/g.cd= lcm

euclid divide krte jao then make the remainder the divisor and divisor the dividened 


code : recursion 
int gcd (int a , int b ){
    if (remainder =O ){
        return divisor ;    // divisor is the gcd 
    }
    // a % b = remainder 
    return gcd(B,a%b)
}