Modulo Function

• Written as % or $\mod x$

The modulo function is the equivalent of taking the remainder of two numbers.

Modular Equivalence

A common way to express modular equivalence by a number $n$ is $x\equiv y\pmod n$

Greatest Common Divisor

• Written as $gcd(x,y)$

The greatest common divisor of two or more integers is the largest positive integer that divides each of the integers.

Least Common Multiple

• Written as $lcm(x,y)$

Euler's Totient Function

• A.k.a. Euler's phi function
• Written as $\varphi(n)$ or $\phi(n)$

Euler's totient function returns the number of positive integers less than $n$ that are [relatively prime]{.spurious-link target=“Greatest Common Divisor”} to $n$.

$\varphi(n)$ is the number of $k\in\mathbb{N}$ such that $k\le n$ and $gcd(k,n)=1$.

Carmichael's Totient Function

• A.k.a. the reduced totient function or the least universal exponent function
• Written as $\lambda(n)$

Carmichael's totient function returns the exponent of the multiplicative group of positive integers modulo $n$.

For every $a\in\mathbb{N}$, $\lambda(n)$ is the smallest $m\in\mathbb{N}$ such that $a^m\equiv1\bmod n$, $a\le n$, and $gcd(a,n)= 1$