This calculator is designed to compute the Euler's Totient function, denoted as \( \phi(n) \), for a given integer \( n \). The Euler's Totient function represents the number of integers up to \( n \) that are coprime (or relatively prime) to \( n \). Two numbers are considered coprime if their greatest common divisor (GCD) is 1, meaning they do not share any common positive factors other than 1.
Not only does this calculator provide the result of \( \phi(n) \), but it also displays step-by-step computations, offering insights into the process. This step-by-step approach can be particularly useful for those learning number theory or for anyone wanting to understand the underlying calculations.
The formula used by this calculator involves identifying the unique prime factors of \( n \) and then applying the formula: \[ \phi(n) = n \prod_{p|n} \left(1 - \frac{1}{p}\right) \] where the product is taken over the distinct prime factors \( p \) of \( n \). This formula provides a methodical way to determine how many integers are coprime to \( n \).
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The Euler's Totient function, \( \phi(n) \), provides a measure of how many integers between 1 and \( n \) are coprime to \( n \). Two numbers are coprime if their greatest common divisor (GCD) is 1.
The power of the Euler's Totient function comes from its relationship with the prime factorization of \( n \). The formula for \( \phi(n) \) is: \begin{equation} \phi(n) = n \prod_{p|n} \left(1 - \frac{1}{p}\right) \end{equation} where the product is taken over the distinct prime numbers \( p \) that divide \( n \). This formula offers an efficient way to compute \( \phi(n) \) without having to individually check each integer up to \( n \).
Example 1: Let's compute \( \phi(12) \). The prime factorization of 12 is \( 2 \times 2 \times 3 \). Using the formula, we identify the unique prime factors as 2 and 3. Thus: \begin{align*} \phi(12) &= 12 \times (1 - \frac{1}{2}) \times (1 - \frac{1}{3}) \\ &= 12 \times \frac{1}{2} \times \frac{2}{3} \\ &= 4 \end{align*} So, there are 4 integers between 1 and 12 that are coprime to 12: 1, 5, 7, and 11.
Example 2: For \( \phi(15) \). The prime factorization of 15 is \( 3 \times 5 \). Using the formula, we get: \begin{align*} \phi(15) &= 15 \times (1 - \frac{1}{3}) \times (1 - \frac{1}{5}) \\ &= 15 \times \frac{2}{3} \times \frac{4}{5} \\ &= 8 \end{align*} The integers between 1 and 15 that are coprime to 15 are: 1, 2, 4, 7, 8, 11, 13, and 14.