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 (altered useful facts section)
(links plurality)
Line 23: Line 23:
 
=== Applications ===
 
=== 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.
+
Modular arithmetic is an extremely useful tool in mathematics competitions. It enables us to easily solve [[Linear diophantine equation]]s, and it often helps with other [[Diophantine equation | Diophantine equations]] as well.
  
  

Revision as of 19:25, 22 June 2006

Modular arithmetic a special type of arithmetic that involves only integers. If two integers ${a},{b}$ leave the same remainder when they are divided by some positive integer ${m}$, we say that ${a}$ and $b$ are congruent modulo ${m}$ or $a\equiv b \pmod {m}$.


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=4012"