How many numbers smaller than 121 are there that are prime to 121?

To determine how many numbers smaller than 121 are prime to 121, we need to find the number of integers less than 121 that are coprime to 121. This is done using **Euler's Totient Function**, denoted as \( \phi(n) \), which counts the number of integers less than \( n \) that are relatively prime to \( n \).

### Steps:
1. **Factorization of 121**:
   \[
   121 = 11^2
   \]
   The prime factor of 121 is 11.

2. **Euler's Totient Function**:
   For \( n = p^k \), where \( p \) is a prime number and \( k \) is a positive integer, Euler's Totient Function is given by:
   \[
   \phi(p^k) = p^k \left( 1 - \frac{1}{p} \right)
   \]

   Applying this to \( 121 = 11^2 \):
   \[
   \phi(121) = 121 \left( 1 - \frac{1}{11} \right) = 121 \times \frac{10}{11} = 110
   \]

### Conclusion:
There are **110 numbers** smaller than 121 that are prime (relatively prime or coprime) to 121.

Would you like more details or further explanations?

Here are 5 related questions you might find useful:
1. What is Euler's Totient Function and how is it calculated for composite numbers?
2. How can we find the prime factorization of a number efficiently?
3. What are the properties of numbers that are coprime to a given number?
4. How does Euler's Totient Function help in cryptography, specifically in RSA encryption?
5. Can Euler's Totient Function be extended to larger numbers with more prime factors?

**Tip**: The Euler's Totient Function is multiplicative, meaning \( \phi(mn) = \phi(m) \times \phi(n) \) when \( m \) and \( n \) are coprime!

Explore how to determine how many numbers smaller than 121 are prime to 121 using Euler's Totient Function. This number theory problem involves coprime numbers and prime factorization. Suitable for high school students, Grades 9-12.

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