What would you like to understand?

To derive $\varphi(p^k)$, we count the integers from 1 to $p^k$ that are coprime to $p^k$. An integer is not coprime to $p^k$ if and only if it shares a prime factor with $p^k$.

Conditions: $p$ is a prime number.; $k$ is a positive integer (implied by context).; The counting range is from 1 to $p^k$ inclusive.

To derive $\varphi(p^k)$, we count the integers from 1 to $p^k$ that are coprime to $p^k$. An integer is not coprime to $p^k$ if and only if it shares a prime factor with $p^k$.

Conditions: $p$ is a prime number.; $k$ is a positive integer (implied by context).; The counting range is from 1 to $p^k$ inclusive.

The general formula for Euler's totient function is $\varphi(n) = n \prod_{i=1}^{m} \left(1 - \frac{1}{p_i}\right)$, where $p_1, p_2, \ldots, p_m$ are the distinct prime factors of $n$. This formula is derived by combining the multiplicativity of $\varphi$, the prime power formula $\varphi(p^k) = p^k - p^{k-1}$, and the Fundamental Theorem of Arithmetic.

Conditions: $n$ is a positive integer greater than 1.; $p_1, \ldots, p_m$ are all the distinct prime factors of $n$.

The general formula for Euler's totient function is $\varphi(n) = n \prod_{i=1}^{m} \left(1 - \frac{1}{p_i}\right)$, where $p_1, p_2, \ldots, p_m$ are the distinct prime factors of $n$. This formula is derived by combining the multiplicativity of $\varphi$, the prime power formula $\varphi(p^k) = p^k - p^{k-1}$, and the Fundamental Theorem of Arithmetic.

Conditions: $n$ is a positive integer greater than 1.; $p_1, \ldots, p_m$ are all the distinct prime factors of $n$.

For a prime number $p$, the only positive divisors are 1 and $p$ itself. Any integer $a$ in the range from 1 to $p$ that is not equal to $p$ will not share the factor $p$.

Conditions: $p$ is a prime number.; $\varphi(p)$ is defined as the count of integers from 1 to $p$ that are coprime to $p$.

For a prime number $p$, the only positive divisors are 1 and $p$ itself. Any integer $a$ in the range from 1 to $p$ that is not equal to $p$ will not share the factor $p$.

Conditions: $p$ is a prime number.; $\varphi(p)$ is defined as the count of integers from 1 to $p$ that are coprime to $p$.

Euler's totient function is multiplicative, meaning $\varphi(nm) = \varphi(n)\varphi(m)$ whenever $\gcd(n, m) = 1$. In the prime factorization of $n$, the factors $p_i^{k_i}$ are powers of distinct primes.

Conditions: $n = p_1^{k_1}p_2^{k_2}\cdots p_m^{k_m}$ is the prime factorization of $n$.; $p_i$ are distinct prime numbers.; $\varphi$ is multiplicative for coprime integers.

Euler's totient function is multiplicative, meaning $\varphi(nm) = \varphi(n)\varphi(m)$ whenever $\gcd(n, m) = 1$. In the prime factorization of $n$, the factors $p_i^{k_i}$ are powers of distinct primes.

Conditions: $n = p_1^{k_1}p_2^{k_2}\cdots p_m^{k_m}$ is the prime factorization of $n$.; $p_i$ are distinct prime numbers.; $\varphi$ is multiplicative for coprime integers.