Order of Element Modulo n Calculator

Discover the order of an element modulo n. This tool helps you find the smallest positive integer k such that a^k ≡ 1 (mod n).

Element (a)

Modulus (n)

Result

The order of modulo is:

Visualization of a^k mod n

k	a^k mod n

What is the Order of an Element Modulo n?

In modular arithmetic, the order of an integer 'a' modulo 'n' is the smallest positive integer 'k' such that a^k is congruent to 1 modulo 'n'. In simpler terms, it's the smallest power 'k' to which you must raise 'a' so that the remainder when a^k is divided by 'n' is 1.

Formula: a^k ≡ 1 (mod n)
Example: The order of 3 modulo 7 is 6, because 3⁶ ≡ 1 (mod 7), and no smaller positive integer exponent works.
Use Cases: Order of elements is a fundamental concept in number theory and has applications in cryptography, especially in algorithms like Diffie-Hellman key exchange.

Learn more about order of elements on resources like Wikipedia and Khan Academy.