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Task/Modular-arithmetic/00-TASK.txt

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+'''[[wp:Modular arithmetic|Modular arithmetic]]''' is a form of arithmetic (a calculation technique involving the concepts of addition and multiplication) which is done on numbers with a defined [[wp:equivalence relation|equivalence relation]] called ''congruence''.  
+
+For any positive integer <math>p</math> called the ''congruence modulus'', 
+two numbers <math>a</math> and <math>b</math> are said to be ''congruent modulo p'' whenever there exists an integer <math>k</math> such that:
+:<math>a = b + k\,p</math>
+
+The corresponding set of [[wp:equivalence class|equivalence class]]es forms a [[wp:ring (mathematics)|ring]] denoted <math>\frac{\Z}{p\Z}</math>.  When p is a prime number, this ring becomes a [[wp:field (mathematics)|field]] denoted <math>\mathbb{F}_p</math>, but you won't have to implement the [[wp:multiplicative inverse|multiplicative inverse]] for this task.
+
+Addition and multiplication on this ring have the same algebraic structure as in usual arithmetic, so that a function such as a polynomial expression could receive a ring element as argument and give a consistent result.
+
+The purpose of this task is to show, if your programming language allows it, 
+how to redefine operators so that they can be used transparently on modular integers.  
+You can do it either by using a dedicated library, or by implementing your own class.
+
+You will use the following function for demonstration:
+:<math>f(x) = x^{100} + x + 1</math>
+You will use <math>13</math> as the congruence modulus and you will compute <math>f(10)</math>.
+
+It is important that the function <math>f</math> is agnostic about whether or not its argument is modular; it should behave the same way with normal and modular integers.  
+In other words, the function is an algebraic expression that could be used with any ring, not just integers.
+<br><br>
+