`(13^(22^(42)))/(9)` find R.

Answer

To find the remainder of \( \frac{13^{22^{42}}}{9} \), we can use the properties of modular arithmetic, specifically Euler's theorem.

### Step-by-Step Solution:

1. **Identify the values**: We need to find \( 13^{22^{42}} \mod 9 \).

2. **Check if 13 and 9 are coprime**: Since the greatest common divisor (gcd) of 13 and 9 is 1, we can apply Euler's theorem.

3. **Calculate Euler's totient function \( \phi(9) \)**:
\[ \phi(9) = 9 \left(1 - \frac{1}{3}\right) = 9 \cdot \frac{2}{3} = 6 \]
Thus, \( \phi(9) = 6 \).

4. **Reduce the exponent \( 22^{42} \mod 6 \)**: We need to calculate \( 22^{42} \mod 6 \).
- First, reduce \( 22 \mod 6 \):
\[ 22 \mod 6 = 4 \]
- Now, we need to compute \( 4^{42} \mod 6 \).

5. **Calculate \( 4^{42} \mod 6 \)**:
- Notice that \( 4 \equiv 4 \mod 6 \) and \( 4^2 \equiv 16 \equiv 4 \mod 6 \).
- Therefore, \( 4^n \equiv 4 \mod 6 \) for any \( n \geq 1 \).
- Thus, \( 4^{42} \mod 6 = 4 \).

6. **Substitute back into the exponent**: Now we have:
\[ 13^{22^{42}} \mod 9 \equiv 13^4 \mod 9 \]

7. **Calculate \( 13^4 \mod 9 \)**:
- First, reduce \( 13 \mod 9 \):
\[ 13 \mod 9 = 4 \]
- Now calculate \( 4^4 \mod 9 \):
\[ 4^2 = 16 \equiv 7 \mod 9 \]
\[ 4^4 = (4^2)^2 = 7^2 = 49 \equiv 4 \mod 9 \]

8. **Final result**: Thus, the remainder when \( 13^{22^{42}} \) is divided by 9 is:
\[ \text{R} = 4 \]

### Summary:
The remainder \( R \) when \( \frac{13^{22^{42}}}{9} \) is calculated is \( 4 \).