# Euler's totient function facts for kids

In number theory, the **totient** of a positive integer *n* is the number of positive integers smaller than *n* which are coprime to *n* (they share no factors except 1). It is often written as .

For example, , because there are four numbers (1, 3, 5 and 7) which do not share any factors with 8. The function used here is the **totient function**, 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 the Greek letter Phi () is so commonly used for it. The **cototient** of *n* is defined as .

The totient function is important mainly because it gives the size of the multiplicative group of integers modulo *n*. More precisely, is the order of the group of units of the ring . This fact, together with Lagrange's theorem, provides a proof for Euler's theorem.

A common use of the totient function is in the RSA algorithm. The RSA algorithm is a popular method of encryption used worldwide.

For any prime number, *p*, .

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