Modulo Function

  • Written as % or modx

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 xy(modn)

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 φ(n) or ϕ(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.

φ(n) is the number of kN such that kn and gcd(k,n)=1.

Carmichael's Totient Function

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

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

For every aN, λ(n) is the smallest mN such that am1modn, an, and gcd(a,n)=1