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**Maths for Programming Contests**

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**Basics Number Represenation in varius bases Basic operations**

Divisibility Reading long long long numbers Basic math functions like factorial

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**GCD of Two Numbers If b|a then gcd(a,b) = b**

Otherwise a = bt+r for some t and r gcd(a,b) = gcd(b,r) gcd(a,b) = gcd(b,a%b) Lcm(a,b) = (a*b)/gcd(a,b) int gcd(int a, int b){ if (b==0) return a; else return gcd(b,a%b); }

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**Prime Numbers Definition Checking if a number is prime**

Generating all prime factors of a number Generating all prime numbers less than a given number (Sieve of Eratosthenes)

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**Modular Arithmetic (x+y) mod n = ((x mod n) + (y mod n))mod n**

(xy) mod n=((xy/2)mod n) ((xy/2)mod n)mod n a simple recursive function to compute the value in O(log n) time.

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**Modular Exponentiation**

Another way to compute it in O(log n) time Y = a0 + a1*2 + a2*22+…+ak*2k X(a+b) = Xa*Xb Xab = (Xa)b XY = Xa0 + (X2)a1 + (X2^2)a2 +….+(X2^k)ak int exp(int x,int y,int mod){ int result=1; while(y){ if((y & 1) == 1){ result = (result*x)%mod; } y = y >> 1; x = (x*x)%mod; return result;

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**Modular Multiplication**

int mulmod(int a,int b,int MOD){ int result = 0,tmp=a,x=b; while(x){ if(x&1){ result += tmp; if(result >=MOD) result -= MOD; } tmp += tmp; if(tmp >= MOD) tmp -= MOD; x = x>>1; return result;

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**Euler’s totient function**

Φ(n) – Number of positive integers less than or equal to n which are coprime to n Φ(ab) = φ(a) φ(b) For a prime p φ(p) = p-1 For a prime p φ(pk) = pk-pk-1 = pk(1-1/p) N = (p1k1)*(p2k2)*…*(pnkn) Φ(N) = (p1k1(1-1/p1) )*…*(pnkn(1-1/pn)) Φ(N) = N(1-1/p1)(1-1/p2)…..(1-1/pn)

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**Sieve technique for Euler’s function**

void prefill(){ int i,j; phi[1] = 0; for(i=2;i<=MAX;i++){ phi[i] = i; } if(phi[i] == i){ for(j=i;j<=MAX;j+=i){ phi[j] = (phi[j]/i)*(i-1);

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**Euler’s Theorem When a and n are co-prime**

if x ≡ y (mod φ(n)), then ax ≡ ay (mod n). aφ(n) ≡ 1 (mod n) (actual theorem the above is a generalization) Euler’s Theorem is a genaralization for Fermat's little theorem

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**Counting Product Rule Sum Rule Principle of inclusion and exclusion**

Closed form expression Recurrence Relations Use of DP to compute the values using the above recurren relations

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**Additional things to look up**

Extended Euclidean algorithm for GCD Chinese Remaindering theorem Mobius Function and its computation using seive

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**Practice http://www.spoj.pl/problems/AU12/**

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