`(5^(500))/(500)` . Find R.

Answer

To find the remainder \( R \) when \( \frac{5^{500}}{500} \) is computed, we can break it down into steps.

### Step 1: Factor the denominator
First, we need to factor \( 500 \):
\[ 500 = 5^3 \times 2^2 \]

### Step 2: Rewrite the expression
Now we can rewrite the expression \( \frac{5^{500}}{500} \) as:
\[ \frac{5^{500}}{5^3 \times 2^2} \]

### Step 3: Simplify the expression
This simplifies to:
\[ \frac{5^{500}}{5^3} \times \frac{1}{2^2} = 5^{500-3} \times \frac{1}{4} = 5^{497} \times \frac{1}{4} \]

### Step 4: Find the remainder when dividing by \( 4 \)
Next, we need to find \( 5^{497} \mod 4 \). Since \( 5 \equiv 1 \mod 4 \), we have:
\[ 5^{497} \equiv 1^{497} \equiv 1 \mod 4 \]

### Step 5: Combine the results
Now, we combine the results:
\[ 5^{497} \times \frac{1}{4} \equiv 1 \times \frac{1}{4} \mod 500 \]
This means that we need to find the remainder of \( 5^{497} \) when divided by \( 500 \).

### Step 6: Calculate \( 5^{497} \mod 500 \)
Since \( 500 = 5^3 \times 2^2 \), we can use the Chinese Remainder Theorem to find \( 5^{497} \mod 500 \).

#### Step 6.1: Calculate \( 5^{497} \mod 125 \)
Since \( 125 = 5^3 \), any power of \( 5 \) greater than or equal to \( 3 \) will be \( 0 \mod 125 \):
\[ 5^{497} \equiv 0 \mod 125 \]

#### Step 6.2: Calculate \( 5^{497} \mod 4 \)
As calculated earlier:
\[ 5^{497} \equiv 1 \mod 4 \]

### Step 7: Solve the system of congruences
Now we have the system of congruences:
\[ x \equiv 0 \mod 125 \]
\[ x \equiv 1 \mod 4 \]

Let \( x = 125k \) for some integer \( k \). Substituting into the second congruence:
\[ 125k \equiv 1 \mod 4 \]
Since \( 125 \equiv 1 \mod 4 \), we have:
\[ k \equiv 1 \mod 4 \]
Thus, \( k = 4m + 1 \) for some integer \( m \).

### Step 8: Substitute back to find \( x \)
Substituting back gives:
\[ x = 125(4m + 1) = 500m + 125 \]
Thus, \( x \equiv 125 \mod 500 \).

### Final Answer
The remainder \( R \) when \( 5^{500} \) is divided by \( 500 \) is:
\[ \boxed{125} \]