6. Verify Euler's Theorem for n = 16 and a = 5. (7 from 6.5)
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![### Verification of Euler's Theorem
**Problem Statement:**
Verify Euler's Theorem for \( n = 16 \) and \( a = 5 \). (7 from 6.5)
Euler's Theorem states that for any integer \( a \) and \( n \) that are coprime (i.e., gcd(a, n) = 1), it holds that:
\[ a^{\phi(n)} \equiv 1 \ (\text{mod} \ n) \]
Here, \( \phi(n) \) is Euler's totient function, which counts the number of integers up to \( n \) that are relatively prime to \( n \).
#### Steps to Verify the Theorem:
1. **Calculate \( \phi(n) \)**:
For \( n = 16 \):
- Since 16 is a power of a prime (i.e., \( 16 = 2^4 \)), we can use the formula for Euler's totient function for powers of primes:
\[ \phi(2^k) = 2^k \left(1 - \frac{1}{2}\right) = 2^k \cdot \frac{1}{2} = 2^{k-1} \]
- Here, \( k = 4 \):
\[ \phi(16) = 2^{4-1} = 2^3 = 8 \]
2. **Verify the modular equivalence**:
For \( a = 5 \):
- We need to check if:
\[ 5^8 \equiv 1 \ (\text{mod} \ 16) \]
Let's calculate \( 5^8 \mod 16 \):
- Calculate \( 5^2 \):
\[ 5^2 = 25 \]
- Find \( 25 \mod 16 \):
\[ 25 \equiv 9 \ (\text{mod} \ 16) \]
- Calculate \( 5^4 \):
\[ 5^4 = (5^2)^2 = 25^2 = 625 \]
- Find \( 625 \mod 16 \):
\[ 625 \mod 16 = 625 - 39 \times 16 = 625 - 624 = 1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffbb66fa7-7c22-4982-a22f-aaed542f65b3%2F9570cd19-8239-4a48-8792-260e6e490c58%2F3g3pghe_processed.png&w=3840&q=75)
Transcribed Image Text:### Verification of Euler's Theorem
**Problem Statement:**
Verify Euler's Theorem for \( n = 16 \) and \( a = 5 \). (7 from 6.5)
Euler's Theorem states that for any integer \( a \) and \( n \) that are coprime (i.e., gcd(a, n) = 1), it holds that:
\[ a^{\phi(n)} \equiv 1 \ (\text{mod} \ n) \]
Here, \( \phi(n) \) is Euler's totient function, which counts the number of integers up to \( n \) that are relatively prime to \( n \).
#### Steps to Verify the Theorem:
1. **Calculate \( \phi(n) \)**:
For \( n = 16 \):
- Since 16 is a power of a prime (i.e., \( 16 = 2^4 \)), we can use the formula for Euler's totient function for powers of primes:
\[ \phi(2^k) = 2^k \left(1 - \frac{1}{2}\right) = 2^k \cdot \frac{1}{2} = 2^{k-1} \]
- Here, \( k = 4 \):
\[ \phi(16) = 2^{4-1} = 2^3 = 8 \]
2. **Verify the modular equivalence**:
For \( a = 5 \):
- We need to check if:
\[ 5^8 \equiv 1 \ (\text{mod} \ 16) \]
Let's calculate \( 5^8 \mod 16 \):
- Calculate \( 5^2 \):
\[ 5^2 = 25 \]
- Find \( 25 \mod 16 \):
\[ 25 \equiv 9 \ (\text{mod} \ 16) \]
- Calculate \( 5^4 \):
\[ 5^4 = (5^2)^2 = 25^2 = 625 \]
- Find \( 625 \mod 16 \):
\[ 625 \mod 16 = 625 - 39 \times 16 = 625 - 624 = 1
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