# Carmichael function

There are two different functions called the **Carmichael function**. Both are similar to Euler's totient function .

## First Definition

The Carmichael function is defined at to be the smallest positive integer such that for all positive integers relatively prime to . The order(group theory) of always divides .

This function is also known as the *reduced totient function* or the *least universal exponent* function.

Suppose . We have

### Examples

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Evaluate . [1]

## Second Definition

The second definition of the Carmichael function is the least common multiples of all the factors of . It is written as . However, in the case , we take as a factor instead of .

### Examples

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