In number theory, the totient p(n) of a positive integer n is defined to be the number of positive integers less than or equal to n that are coprime to n. For example, p(9) = 6 since the six numbers 1, 2, 4, 5, 7 and 8 are coprime to 9. The function p so defined is the totient function. The totient is usually called the Euler totient or Euler's totient, after the Swiss mathematician Leonhard Euler, who studied it. The totient function is also called Euler's phi function or simply the phi function, since it is commonly denoted by the Greek letter Phi (p). The cototient of n is defined as n - p(n); the number of positive integers less than or equal to n that are not coprime to n.
The totient function is important mainly because it gives the size of the multiplicative group of integers modulo n. More precisely, p(n) is the order of the group of units of the ring Z/nZ. This fact, together with Lagrange's theorem, provides a proof for Euler's theorem.
Reference:
http://en.wikipedia.org/wiki/Euler%27s_totient_function
