The Euler's Totient function $\varphi(n)$, denoted as $\varphi(n)$, counts the number of integers from 1 to $n$ that are relatively prime to $n$. The formula for the totient function $\varphi(n)$ for a number $n$ is given by:

$\varphi(n) = n \times \prod_{p | n} \left(1 - \frac{1}{p}\right)$

Where $p$ are the distinct prime factors of $n$.

Let's calculate each one step-by-step:

1. $\varphi(28)$

The prime factorization of 28 is $28 = 2^2 \times 7$.

Using the totient formula:

= 28 \times \frac{1}{2} \times \frac{6}{7} = 12$$ So, $$\varphi(28) = 12$$. #### 2. $$\varphi(41)$$ Since 41 is a prime number, the totient function for any prime $$p$$ is: $$\varphi(41) = 41 - 1 = 40$$ So, $$\varphi(41) = 40$$. #### 3. $$\varphi(231)$$ The prime factorization of 231 is $$231 = 3 \times 7 \times 11$$. Using the totient formula: $$\varphi(231) = 231 \times \left(1 - \frac{1}{3}\right) \times \left(1 - \frac{1}{7}\right) \times \left(1 - \frac{1}{11}\right)$$ $$\varphi(231) = 231 \times \frac{2}{3} \times \frac{6}{7} \times \frac{10}{11} = 120$$ So, $$\varphi(231) = 120$$. #### 4. $$\varphi(440)$$ The prime factorization of 440 is $$440 = 2^3 \times 5 \times 11$$. Using the totient formula: $$\varphi(440) = 440 \times \left(1 - \frac{1}{2}\right) \times \left(1 - \frac{1}{5}\right) \times \left(1 - \frac{1}{11}\right)$$ $$\varphi(440) = 440 \times \frac{1}{2} \times \frac{4}{5} \times \frac{10}{11} = 160$$ So, $$\varphi(440) = 160$$. ### Final answers: - $$\varphi(28) = 12$$ - $$\varphi(41) = 40$$ - $$\varphi(231) = 120$$ - $$\varphi(440) = 160$$ Would you like any detailed explanation on a specific part? Here are 5 related questions: 1. What is Euler’s Totient function for prime powers like $$\varphi(2^n)$$? 2. How is Euler’s Totient function related to cryptography, particularly in RSA? 3. What are the properties of the Totient function for composite numbers? 4. How does the Euler’s Totient function behave for large prime numbers? 5. Can Euler's Totient function be extended to non-integers? **Tip:** The Euler's Totient function is especially important in number theory and is frequently used in problems involving modular arithmetic and cryptography.