Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Modular arithmetic"

m
Line 1: Line 1:
 
'''Modular arithmetic''' is a special type of arithmetic that involves only [[integers]].  Given integers <math>a</math>, <math>b</math>, and <math>n</math>, with <math>n > 0</math>, we say that <math>a</math> is ''congruent to'' <math>b</math> ''modulo'' <math>n</math>, or <math>a \equiv b</math> (mod <math>n</math>), if the difference <math>{a - b}</math> is divisible by <math>n</math>.
 
'''Modular arithmetic''' is a special type of arithmetic that involves only [[integers]].  Given integers <math>a</math>, <math>b</math>, and <math>n</math>, with <math>n > 0</math>, we say that <math>a</math> is ''congruent to'' <math>b</math> ''modulo'' <math>n</math>, or <math>a \equiv b</math> (mod <math>n</math>), if the difference <math>{a - b}</math> is divisible by <math>n</math>.
  
−
For a given positive integer <math>n</math>, the relation <math>a \equiv b</math> (mod <math>n</math>) is an [[equivalence relation]] on the set of integers.  This relation gives rise to an algebraic structure called '''the integers modulo <math>n</math>''' (usually known as "the integers mod <math>n</math>," or <math>\mathbb{Z}_n</math> for short).  This structure gives us a useful tool for solving a wide range of number-theoretic problems, including finding solutions to [[Diophantine equation]]s, testing whether certain large numbers are prime, and even some problems in cryptology.
+
For a given positive integer <math>n</math>, the relation <math>a \equiv b</math> (mod <math>n</math>) is an [[equivalence relation]] on the set of integers.  This relation gives rise to an algebraic structure called '''the integers modulo <math>n</math>''' (usually known as "the integers mod <math>n</math>," or <math>\mathbb{Z}_n</math> for short).  This structure gives us a useful tool for solving a wide range of number-theoretic problems, including finding solutions to [[Diophantine equation|Diophantine equations]], testing whether certain large numbers are prime, and even some problems in cryptology.
  
  

Revision as of 20:55, 23 June 2006

Modular arithmetic is a special type of arithmetic that involves only integers. Given integers $a$, $b$, and $n$, with $n > 0$, we say that $a$ is congruent to $b$ modulo $n$, or $a \equiv b$ (mod $n$), if the difference ${a - b}$ is divisible by $n$.

For a given positive integer $n$, the relation $a \equiv b$ (mod $n$) is an equivalence relation on the set of integers. This relation gives rise to an algebraic structure called the integers modulo $n$ (usually known as "the integers mod $n$," or $\mathbb{Z}_n$ for short). This structure gives us a useful tool for solving a wide range of number-theoretic problems, including finding solutions to Diophantine equations, testing whether certain large numbers are prime, and even some problems in cryptology.


Introductory

Useful Facts

Consider four integers ${a},{b},{c},{d}$ and a positive integer ${m}$ such that $a\equiv b\pmod {m}$ and $c\equiv d\pmod {m}$. In modular arithmetic, the following identities hold:

  • Addition: $a+c\equiv b+d\pmod {m}$.
  • Substraction: $a-c\equiv b-d\pmod {m}$.
  • Multiplication: $ac\equiv bd\pmod {m}$.
  • Division: $\frac{a}{e}\equiv \frac{b}{e}\pmod {\frac{m}{\gcd(m,e)}}$, where $e$ is a positive integer that divides ${a}$ and $b$.
  • Exponentiation: $a^e\equiv b^e\pmod {m}$ where $e$ is a positive integer.

Examples

  • ${7}\equiv {1} \pmod {2}$
  • $49^2\equiv 7^4\equiv (1)^4\equiv 1 \pmod {6}$
  • $7a\equiv 14\pmod {49}\implies a\equiv 2\pmod {7}$

Applications

Modular arithmetic is an extremely useful tool in mathematics competitions. It enables us to easily solve Linear Diophantine equations, and it often helps with other Diophantine equations as well.

Intermediate

Topics

See also

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Modular_arithmetic&oldid=4394"