`(25^(29))/(28)` find R.

Answer

To find the remainder \( R \) when \( \frac{25^{29}}{28} \) is calculated, we can follow these steps:

### Step 1: Rewrite 25 in terms of 28
We can express 25 as \( 28 - 3 \):
\[ 25 \equiv -3 \mod 28 \]

### Step 2: Substitute into the expression
Now, we can substitute \( 25 \) in the expression:
\[ 25^{29} \equiv (-3)^{29} \mod 28 \]

### Step 3: Calculate \( (-3)^{29} \)
Since \( (-3)^{29} = -3^{29} \), we need to compute \( 3^{29} \mod 28 \).

### Step 4: Simplify \( 3^{29} \mod 28 \)
To simplify \( 3^{29} \mod 28 \), we can use properties of modular arithmetic. We can calculate the powers of 3 modulo 28:
- \( 3^1 \equiv 3 \)
- \( 3^2 \equiv 9 \)
- \( 3^3 \equiv 27 \equiv -1 \mod 28 \)

### Step 5: Use the result from \( 3^3 \)
Since \( 3^3 \equiv -1 \mod 28 \), we can express \( 3^{29} \) in terms of \( 3^3 \):
\[ 3^{29} = (3^3)^9 \cdot 3^2 \equiv (-1)^9 \cdot 9 \equiv -9 \mod 28 \]

### Step 6: Substitute back into the expression
Now substituting back, we have:
\[ (-3)^{29} \equiv -(-9) \equiv 9 \mod 28 \]

### Step 7: Find the final remainder
Thus, the remainder \( R \) when \( 25^{29} \) is divided by 28 is:
\[ R = 9 \]

### Final Answer
The remainder \( R \) is \( 9 \).

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