Using Euler’s theorem calculate 5^123 in Z33

To calculate \( 5^{123} \mod 33 \) using **Euler's theorem**, follow these steps:

### Step 1: Euler's Theorem Overview
Euler's theorem states that if \( a \) and \( n \) are coprime (i.e., \( \gcd(a, n) = 1 \)), then:
\[
a^{\phi(n)} \equiv 1 \mod n
\]
where \( \phi(n) \) is Euler's totient function of \( n \).

### Step 2: Check if 5 and 33 are Coprime
Since \( \gcd(5, 33) = 1 \) (they have no common factors), Euler's theorem can be applied.

### Step 3: Compute \( \phi(33) \)
Euler’s totient function \( \phi(n) \) is calculated as:
\[
\phi(33) = \phi(3 \times 11) = \phi(3) \times \phi(11)
\]
where:
- \( \phi(3) = 3 - 1 = 2 \)
- \( \phi(11) = 11 - 1 = 10 \)

So, \( \phi(33) = 2 \times 10 = 20 \).

### Step 4: Reduce Exponent Using Euler's Theorem
From Euler's theorem, we know:
\[
5^{20} \equiv 1 \mod 33
\]
Now reduce the exponent \( 123 \mod 20 \):
\[
123 \div 20 = 6 \text{ remainder } 3
\]
Thus:
\[
5^{123} \equiv 5^3 \mod 33
\]

### Step 5: Calculate \( 5^3 \mod 33 \)
Now calculate \( 5^3 \):
\[
5^3 = 5 \times 5 \times 5 = 125
\]
Now reduce \( 125 \mod 33 \):
\[
125 \div 33 = 3 \text{ remainder } 26
\]
Thus:
\[
125 \equiv 26 \mod 33
\]

### Final Answer:
\[
5^{123} \equiv 26 \mod 33
\]

Would you like more details or have any questions?

Here are some related questions:
1. How do you compute \( \phi(n) \) for larger numbers?
2. How does Euler's theorem relate to Fermat's Little Theorem?
3. Can Euler’s theorem be used if \( a \) and \( n \) are not coprime?
4. What are other applications of Euler’s theorem in cryptography?
5. How do you calculate powers efficiently using modular arithmetic?

**Tip:** Modular exponentiation can be computed efficiently using the method of "exponentiation by squaring," especially for large exponents.

Learn how to use Euler’s theorem to calculate 5^123 mod 33 step-by-step. This problem involves number theory concepts, including modular arithmetic, the totient function, and Euler's theorem. Suitable for advanced high school or early college students.

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