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Number theory Part-2

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Introduction Problems in competitive programming which involve Mathematics are are usually about number theory, or geometry. If you know number theory, that increases your ammo heavily in solving a lot of tougher problems, and helps you in getting a strong hold on a lot of other problems, too. Problems in competitive programming require insight, so just knowing some topics is not enough at all. All of the problems requires more or less math tough. For instance, solving large systems of equations and approximating solutions to differential equations.  Set Theory Before we proceed, let us get through the some common set operations. Q.) What is a Set? -In mathematics, a set is a collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written {2,4,6}. Sets are one of the most fundamental con...
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make pair usuage

The behavior of this function template is the same as if defined as: 1 2 3 4 5 template < class T1, class T2> pair<T1,T2> make_pair (T1 x, T2 y) { return ( pair<T1,T2>(x,y) ); } // make_pair example #include <utility> // std::pair #include <iostream> // std::cout int main () { std::pair < int , int > foo; std::pair < int , int > bar; foo = std::make_pair (10,20); bar = std::make_pair (10.5, 'A' ); // ok: implicit conversion from pair<double,char> std::cout << "foo: " << foo.first << ", " << foo.second << '\n' ; std::cout << "bar: " << bar.first << ", " << bar.second << '\n' ; return 0; } The difference is that with  std::pair  you need to specify the types of both elements, whereas  std::make_pair  will create a pair with the type of the elements that are ...
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Number Theory Part-1

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Introduction: Problems in competitive programming which involve Mathematics are are usually about number theory, or geometry. If you know number theory, that increases your ammo heavily in solving a lot of tougher problems, and helps you in getting a strong hold on a lot of other problems, too. Problems in competitive programming require insight, so just knowing some topics is not enough at all. All of the problems requires more or less math tough. For instance, solving large systems of equations and approximating solutions to differential equations. Modulo: Modulo operation gives the remainder after division, when one number is divided by another. It is denoted by % sign. Example: We have two numbers 5 and 2, then 5%2 is 1 as when 5 is divided by 2, it leaves 1 as remainder. Properties: Some of the properties of modulo are: (a+b)%c = (a%c + b%c )%c. (a * b)%c = ((a%c) * (b%c))%c. Example: Let’s say a = 5, b = 3, c = 2. Then: 1) (5+3)%2 = 8%2 = 0. Similarly (5%2 ...
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