`(25^(26^(27)))/(11)` Find R.

Answer

To solve the problem \((25^{(26^{27})}) / 11\) and find the remainder \(R\), we can use the properties of modular arithmetic and the Tochant method. Here’s a step-by-step solution:

### Step 1: Identify Co-primality
First, we need to check if 25 and 11 are co-prime. Since the greatest common divisor (GCD) of 25 and 11 is 1, they are co-prime.

**Hint:** Two numbers are co-prime if their GCD is 1.

### Step 2: Calculate \(T\)
Using the Tochant method, we calculate \(T\) as follows:
\[ T = 11 \times \left(1 - \frac{1}{11}\right) = 11 \times \frac{10}{11} = 10 \]

**Hint:** The formula for \(T\) in the Tochant method is \(d \times \left(1 - \frac{1}{d}\right)\).

### Step 3: Reduce the Exponent Modulo \(T\)
Next, we need to find \(26^{27} \mod 10\):
- Calculate \(26 \mod 10 = 6\).
- Therefore, we need to find \(6^{27} \mod 10\).

**Hint:** Reducing the base modulo \(T\) simplifies the exponentiation.

### Step 4: Find \(6^{27} \mod 10\)
To find \(6^{27} \mod 10\), we observe the pattern of powers of 6 modulo 10:
- \(6^1 \equiv 6 \mod 10\)
- \(6^2 \equiv 6 \mod 10\)
- \(6^3 \equiv 6 \mod 10\)
- ...

Thus, \(6^n \equiv 6 \mod 10\) for any \(n \geq 1\).

So, \(6^{27} \equiv 6 \mod 10\).

**Hint:** Recognizing patterns in powers can simplify calculations.

### Step 5: Substitute Back into the Original Expression
Now we substitute back:
\[ 25^{(26^{27})} \equiv 25^6 \mod 11 \]

**Hint:** We can reduce the exponent using the result from the previous step.

### Step 6: Calculate \(25^6 \mod 11\)
First, reduce \(25 \mod 11\):
\[ 25 \equiv 3 \mod 11 \]
Thus, we need to calculate \(3^6 \mod 11\).

**Hint:** Always reduce the base modulo \(d\) before exponentiation.

### Step 7: Calculate \(3^6 \mod 11\)
We can compute \(3^6\) as follows:
- \(3^2 = 9\)
- \(3^4 = 9^2 = 81 \equiv 4 \mod 11\)
- \(3^6 = 3^4 \cdot 3^2 \equiv 4 \cdot 9 = 36 \equiv 3 \mod 11\)

**Hint:** Break down the exponentiation into smaller parts to simplify calculations.

### Step 8: Conclusion
The remainder when \((25^{(26^{27})})\) is divided by 11 is:
\[ R = 3 \]

### Final Answer
Thus, the final answer is:
\[ \boxed{3} \]